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Palindromic Decremented Squares
of the form n^2–x
rood n(n+0) rood n(n+1) rood n(n+2) rood n(n+x) rood
rood n^2+1 rood n^2+x rood
rood n^2+(n+1) rood n^2+(n+x)



Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random examples : 7, 3113, 44611644

Decremented Square numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base2 – X

     PLAIN TEXT SQUDECN 

Palindromic Decremented Squares

Decremented Squares of the form x2 – 0 can only start or end with a 0, 1, 4, 5, 6 or 9
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can only be followed by an even number : 10, 12, 14, 16 or 18
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can only be followed by 2 : 52
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
9 can only be followed by an even number : 90, 92, 94, 96 or 98

Decremented Squares of the form x2 – 1 can only start or end in one of the following digits : 0, 3, 4, 5, 8 or 9.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.
3 can only be followed by an even number : 30, 32, 34, 36 or 38
4 can only be followed by 2 : 42
5 can only be followed by an odd number : 51, 53, 55, 57, 59
8 can only be followed by an even number : 80, 82, 84, 86 or 88
9 can only be followed by 9 : 99
Also Palindromic Quasipronics of the form x*(x + 2).
Proof by substitution whereby x equals  n – 1 
x * ( x + 2 ) =
      (n – 1) * ( (n – 1) + 2 ) =
      (n – 1) * (n + 1) =
      n2 + n –n – 1 =
n2 – 1 [ QED ]

Decremented Squares of the form x2 – 2 can only start or end with a 2, 3, 4, 7, 8 or 9.
Even length palindromes of this form are not possible as they are never divisible by 11.
2 can only be followed by an even number : 20, 22, 24, 26 or 28
3 can only be followed by 2 : 32
4 can only be followed by an odd number : 41, 43, 45, 47 or 49
7 can only be followed by an even number : 70, 72, 74, 76 or 78
8 can only be followed by 9 : 89
9 can only be followed by an odd number : 91, 93, 95, 97 or 99

Decremented Squares of the form x2 – 3 can only start or end with a 1, 2, 3, 6, 7 or 8.
1 can only be followed by an even number : 10, 12, 14, 16 or 18
2 can only be followed by 2 : 22
3 can only be followed by an odd number : 31, 33, 35, 37 or 39
6 can only be followed by an even number : 60, 62, 64, 66 or 68
7 can only be followed by 9 : 79
8 can only be followed by an odd number : 81, 83, 85, 87 or 89

Decremented Squares of the form x2 – 4 can only end with a 0, 1, 2, 5, 6 or 7.
And if it terminates in 1, it terminates in 21; if it terminates in 6, it terminates in 96 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.
1 can only be followed by 2 : 12
2 can only be followed by an odd number : 21, 23, 25, 27 or 29
5 can only be followed by an even number : 50, 52, 54, 56 or 58
6 can only be followed by 9 : 69
7 can only be followed by an odd number : 71, 73, 75, 77 or 79
Also Palindromic Quasipronics of the form x*(x + 4).
Proof by substitution whereby x equals  n – 2 
x * ( x + 4 ) =
      (n – 2) * ( (n – 2) + 4 ) =
      (n – 2) * (n + 2) =
      n2 + 2n –2n – 4 =
n2 – 4 [ QED ]

Decremented Squares of the form x2 – 5 can only start or end with a 0, 1, 4, 5, 6 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can only be followed by an odd number : 11, 13, 15, 17 or 19
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can only be followed by 9 : 59
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
9 can only be followed by an odd number : 91, 93, 95, 97 or 99

Decremented Squares of the form x2 – 6 can only start or end with a 0, 3, 4, 5, 8 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
Even length palindromes of this form are not possible as they are never divisible by 11.
3 can only be followed by an even number : 30, 32, 34, 36 or 38 { This 36... produces ever expanding Mandelbrot-like infinite palindromes }
4 can only be followed by 9 : 49
5 can only be followed by an odd number : 51, 53, 55, 57 or 59
8 can only be followed by an odd number : 81, 83, 85, 87 or 89
9 can only be followed by 1 : 91

Decremented Squares of the form x2 – 7 can only start or end with a 2, 3, 4, 7, 8 or 9.
Even length palindromes of this form are not possible as they are never divisible by 11.
2 can only be followed by an even number : 20, 22, 24, 26 or 28
3 can only be followed by 9 : 39
4 can only be followed by an odd number : 41, 43, 45, 47 or 49
7 can only be followed by an odd number : 71, 73, 75, 77 or 79 { This 77... produces ever expanding Mandelbrot-like infinite palindromes }
8 can only be followed by 1 : 81
9 can only be followed by an even number : 90, 92, 94, 96 or 98

Decremented Squares of the form x2 – 8 can only start or end with a 1, 2, 3, 6, 7 or 8.
Even length palindromes of this form are not possible as they are never divisible by 11.
1 can only be followed by an even number : 10, 12, 14, 16 or 18
2 can only be followed by 9 : 29
3 can only be followed by an odd number : 31, 33, 35, 37 or 39
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
7 can only be followed by 1 : 71
8 can only be followed by an even number : 80, 82, 84, 86 or 88

Decremented Squares of the form x2 – 9 can only start or end with a 0, 1, 2, 5, 6 or 7.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can only be followed by 9 : 19
2 can only be followed by an odd number : 21, 23, 25, 27 or 29
5 can only be followed by an odd number : 51, 53, 55, 57 or 59
6 can only be followed by 1 : 61
7 can only be followed by an even number : 70, 72, 74, 76 or 78
Also Palindromic Quasipronics of the form x*(x + 6).
Proof by substitution whereby x equals  n – 3 
x * ( x + 6 ) =
      (n – 3) * ( (n – 3) + 6 ) =
      (n – 3) * (n + 3) =
      n2 + 3n –3n – 9 =
n2 – 9 [ QED ]


An infinite palindromic pattern hides in the list for case n^2 – 2

302 2 = 898
3002 2 = 89998
30002 2 = 8999998
300002 2 = 899999998
3000002 2 = 89999999998
30000002 2 = 8999999999998
300000002 2 = 899999999999998

Two infinite palindromic pattern hide in the list for case n^2 – 3

292 3 = 838
2992 3 = 89398
29992 3 = 8993998
299992 3 = 899939998
2999992 3 = 89999399998
29999992 3 = 8999993999998
299999992 3 = 899999939999998
122 3 = 141
1022 3 = 10401
10022 3 = 1004001
100022 3 = 100040001
1000022 3 = 10000400001
10000022 3 = 1000004000001
100000022 3 = 100000040000001

One finite palindromic pattern starting from length 15 !

281111102 3 = 790234505432097
2811111102 3 = 79023456165432097
28111111102 3 = 7902345672765432097
281111111102 3 = 790234567838765432097
2811111111102 3 = 79023456789498765432097

Four infinite palindromic pattern (probably more) hides in the list for case n^2 – 6

19132 6 = 3659563
190132 6 = 361494163
1900132 6 = 36104940163
19000132 6 = 3610049400163
190000132 6 = 361000494000163
1900000132 6 = 36100004940000163
1910872 6 = 36514241563
190109872 6 = 361417626714163
19001099872 6 = 3610417962697140163
1900010999872 6 = 36100417996269971400163
190000109999872 6 = 361000417999626999714000163
19000001099999872 6 = 3610000417999962699997140000163

This infinite palindromic pattern clean starting from length 19 !

19108913 [365150556041563] {L=15} Alas! Not palindromic but a NEAR MISS
19010890132 6 = 3614139435349314163 {L=19}
1900108900132 6 = 36104138323532383140163 {L=23}
190001089000132 6 = 361004138212353212831400163 {L=27}
19000010890000132 6 = 3610004138201235321028314000163 {L=31}
1900000108900000132 6 = 36100004138200123532100283140000163 {L=35}
190000001089000000132 6 = 361000004138200012353210002831400000163 {L=39}

This infinite palindromic pattern clean starting from length 25 !

1910989132 6 = 36518794549781563 {L=17}
19010989013 [361417703252406714163] {L=21} Alas! Not palindromic but a NEAR MISS
19001098900132 6 = 3610417594125214957140163 {L=25}
1900010989000132 6 = 36100417583212521238571400163 {L=29}
190000109890000132 6 = 361000417582121252121285714000163 {L=33}
19000001098900000132 6 = 3610000417582012125212102857140000163 {L=37}
1900000010989000000132 6 = 36100000417582001212521210028571400000163 {L=41}
190000000109890000000132 6 = 361000000417582000121252121000285714000000163 {L=45}
19000000001098900000000132 6 = 3610000000417582000012125212100002857140000000163 {L=49}
1900000000010989000000000132 6 = 36100000000417582000001212521210000028571400000000163 {L=53}

This finite (triplet) palindromic pattern clean starting from length 23 !

1901089109872 6 = 36141398036663089314163 {L=23}
190010891099872 6 = 361041387365666563783140163 {L=27}
19000108910999872 6 = 3610041386298566658926831400163 {L=31}

An infinite palindromic pattern hides in the list for case n^2 – 7

202 7 = 393
2002 7 = 39993
20002 7 = 3999993
200002 7 = 399999993
2000002 7 = 39999999993
20000002 7 = 3999999999993
200000002 7 = 399999999999993

A finite palindromic pattern with only four steps

1(63)2 7 = 26562
1(63)(63)2 7 = 267747762
1(63)(63)(63)2 7 = 2677683867762
1(63)(63)(63)(63)2 7 = 26776859295867762

Two infinite palindromic pattern hide in the list for case n^2 – 8

192 8 = 353
1992 8 = 39593
19992 8 = 3995993
199992 8 = 399959993
1999992 8 = 39999599993
19999992 8 = 3999995999993
199999992 8 = 399999959999993
132 8 = 161
1032 8 = 10601
10032 8 = 1006001
100032 8 = 100060001
1000032 8 = 10000600001
10000032 8 = 1000006000001
100000032 8 = 100000060000001


The Table


Index Nr BasenumberLength
Palindromic Decremented Squares of form n^2–XLength
up down
Case X = 0
See in-depth webpage about Palindromic Square Numbers
up down
Case X = 1
See in-depth webpage about Palindromic Quasipronic Numbers of the form n(n+2)
up down
Case X = 2
One can find the regular numbers of the form n2 – 2 at
%N n2 – 2. under A059100.
The palindromic numbers of the form n2 – 2 are categorised as follows :
%N n2 – 2 is a palindrome. under A0?????.
%N Palindromes of the form n2 – 2. under A0?????.
5630.000.000.000.000.000.00020
899.999.999.999.999.999.999.999.999.999.999.999.99839
5527.320.258.176.172.852.40720
746.396.506.812.739.591.747.195.937.218.605.693.64739
543.000.000.000.000.000.00019
8.999.999.999.999.999.999.999.999.999.999.999.99837
532.178.015.454.901.896.42619
4.743.751.321.791.514.824.284.151.971.231.573.47437
521.509.542.387.634.560.58219
2.278.718.220.065.449.960.699.445.600.228.178.72237
51300.000.000.000.000.00018
89.999.999.999.999.999.999.999.999.999.999.99835
50277.046.141.819.110.63718
76.754.564.696.854.763.736.745.869.646.545.76735
49167.724.793.753.032.07218
28.131.606.439.497.146.364.179.493.460.613.18235
48163.836.377.167.092.80818
26.842.358.483.237.889.598.873.238.485.324.86235
4730.000.000.000.000.00017
899.999.999.999.999.999.999.999.999.999.99833
4626.864.628.003.947.87717
721.708.237.790.500.494.005.097.732.807.12733
453.057.109.285.491.27116
9.345.917.183.436.949.496.343.817.195.43931
443.000.000.000.000.00016
8.999.999.999.999.999.999.999.999.999.99831
432.997.152.982.455.77016
8.982.926.000.243.517.153.420.006.292.89831
42300.000.000.000.00015
89.999.999.999.999.999.999.999.999.99829
41151.051.744.377.76815
22.816.629.479.566.566.597.492.661.82229
4030.000.000.000.00014
899.999.999.999.999.999.999.999.99827
393.000.000.000.00013
8.999.999.999.999.999.999.999.99825
382.736.719.294.29313
7.489.632.495.755.575.942.369.84725
371.434.419.498.75213
2.057.559.298.399.938.929.557.50225
36300.000.000.00012
89.999.999.999.999.999.999.99823
35277.104.474.86312
76.786.889.989.098.998.868.76723
34169.944.971.37812
28.881.293.296.669.239.218.88223
33150.408.075.56812
22.622.589.196.069.198.522.62223
3230.000.000.00011
899.999.999.999.999.999.99821
3120.470.241.05411
419.030.768.808.867.030.91421
3020.414.740.29611
416.761.621.353.126.167.61421
2920.371.679.75411
415.005.335.999.533.500.51421
283.000.000.00010
8.999.999.999.999.999.99819
272.759.060.76310
7.612.416.293.926.142.16719
262.728.091.60710
7.442.483.816.183.842.44719
251.807.133.72510
3.265.732.300.032.375.62319
241.489.076.31810
2.217.348.280.828.437.12219
23314.838.8999
99.123.532.323.532.19917
22300.000.0009
89.999.999.999.999.99817
2130.000.0008
899.999.999.999.99815
2020.367.6968
414.843.040.348.41415
193.000.0007
8.999.999.999.99813
18300.0006
89.999.999.99811
17202.5046
41.007.870.01411
16151.2326
22.871.117.82211
15144.2986
20.821.912.80211
1430.0005
899.999.9989
133.1494
Prime!    9.916.1997
123.0004
8.999.9987
112.1744
4.726.2747
102.0944
4.384.8347
93003
89.9985
82043
41.6145
7Prime!    312
9593
6302
8983
5272
Prime!    7273
4Prime!    31
Prime!    71
3Prime!    21
Prime!    21
211
–11
101
–21
up down
Case X = 3
One can find the regular numbers of the form n2 – 3 at
%N n2 – 3. under A028872.
The palindromic numbers of the form n2 – 3 are categorised as follows :
%N n2 – 3 is a palindrome. under A0?????.
%N Palindromes of the form n2 – 3. under A0?????.
114800.096.871.932.978.49318
640.155.004.476.936.987.789.639.674.400.551.04636
113432.342.232.793.964.32218
186.919.806.257.270.438.834.072.752.608.919.68136
112299.999.999.999.999.99918
89.999.999.999.999.999.399.999.999.999.999.99835
111250.947.643.580.916.82718
62.974.719.818.814.866.366.841.881.891.747.92635
110129.576.704.573.474.54218
16.790.122.368.121.498.289.412.186.322.109.76135
109112.894.940.866.782.08218
12.745.267.673.314.223.032.241.337.676.254.72135
108100.000.000.000.000.00218
10.000.000.000.000.000.400.000.000.000.000.00135
10737.831.729.891.151.31217
1.431.239.786.557.031.661.307.556.879.321.34134
10629.999.999.999.999.99917
899.999.999.999.999.939.999.999.999.999.99833
10510.000.000.000.000.00217
100.000.000.000.000.040.000.000.000.000.00133
1043.813.173.873.483.11216
14.540.294.989.414.200.241.498.949.204.54132
1033.225.129.724.100.10216
10.401.461.737.274.000.047.273.716.410.40132
1023.213.817.208.205.04816
10.328.621.047.754.888.845.774.012.682.30132
1012.999.999.999.999.99916
8.999.999.999.999.993.999.999.999.999.99831
1002.859.912.361.666.81116
8.179.098.716.414.636.364.146.178.909.71831
991.984.826.623.628.58616
3.939.536.725.864.852.584.685.276.359.39331
981.975.077.481.649.56416
3.900.931.058.519.183.819.158.501.390.09331
971.363.557.307.075.17216
1.859.288.529.678.094.908.769.258.829.58131
961.104.380.147.463.31816
1.219.655.510.111.100.011.110.155.569.12131
951.000.000.000.000.00216
1.000.000.000.000.004.000.000.000.000.00131
94925.076.769.015.58115
855.767.028.572.306.603.275.820.767.55830
93793.326.228.722.92315
629.366.505.179.735.537.971.505.663.92630
92299.999.999.999.99915
89.999.999.999.999.399.999.999.999.99829
91257.617.848.652.46315
66.366.955.944.323.332.344.955.966.36629
90187.685.767.311.01615
35.225.947.251.124.842.115.274.952.25329
89177.196.446.108.35415
31.398.580.513.430.803.431.508.589.31329
88111.009.388.950.48215
12.323.084.435.159.395.153.448.032.32129
87100.000.000.000.00215
10.000.000.000.000.400.000.000.000.00129
8629.999.999.999.99914
899.999.999.999.939.999.999.999.99827
8528.927.607.042.87914
836.806.449.227.222.722.944.608.63827
8418.401.474.433.90614
338.614.261.341.696.143.162.416.83327
8315.052.520.152.52514
226.578.362.942.171.249.263.875.62227
8214.843.663.905.69514
220.334.358.145.232.541.853.433.02227
8111.982.083.209.81214
143.570.318.046.858.640.813.075.34127
8010.000.000.000.00214
100.000.000.000.040.000.000.000.00127
796.108.627.042.87613
37.315.324.348.955.984.342.351.37326
784.700.308.259.34513
22.092.897.732.866.823.779.829.02226
772.999.999.999.99913
8.999.999.999.993.999.999.999.99825
762.542.454.379.84313
6.464.074.273.582.853.724.704.64625
751.108.553.389.83213
1.228.890.618.108.018.160.988.22125
741.000.000.000.00213
1.000.000.000.004.000.000.000.00125
73299.999.999.99912
89.999.999.999.399.999.999.99823
72298.530.039.85112
89.120.184.693.439.648.102.19823
71293.254.640.78112
85.998.284.339.593.348.289.95823
70281.111.111.11012
79.023.456.789.498.765.432.09723
69149.088.589.58512
22.227.407.544.444.570.472.22223
68134.952.824.27812
Prime!    18.212.264.780.608.746.221.28123
67100.000.000.00212
10.000.000.000.400.000.000.00123
6638.318.240.26211
1.468.287.536.776.357.828.64122
6529.999.999.99911
899.999.999.939.999.999.99821
6428.111.111.11011
790.234.567.838.765.432.09721
6310.145.537.69811
102.931.935.181.539.139.20121
6210.000.000.00211
100.000.000.040.000.000.00121
615.748.742.99410
33.048.046.011.064.084.03320
602.999.999.99910
8.999.999.993.999.999.99819
592.811.111.11010
7.902.345.672.765.432.09719
582.578.854.58710
6.650.490.980.890.940.56619
571.485.275.39510
2.206.042.998.992.406.02219
561.104.201.68210
Prime!    1.219.261.354.531.629.12119
551.038.329.05210
1.078.127.220.227.218.70119
541.000.000.00210
1.000.000.004.000.000.00119
53299.999.9999
89.999.999.399.999.99817
52281.111.1109
79.023.456.165.432.09717
51198.699.8649
39.481.635.953.618.49317
50119.193.8389
14.207.171.017.170.24117
49100.000.0029
10.000.000.400.000.00117
4892.274.8818
8.514.653.663.564.15816
4759.963.8668
3.595.665.225.665.95316
4643.535.7728
1.895.363.443.635.98116
45Prime!    29.999.9998
899.999.939.999.99815
4428.111.1108
790.234.505.432.09715
4324.544.2978
602.422.515.224.20615
4218.750.5348
351.582.525.285.15315
4110.000.0028
100.000.040.000.00115
409.260.0697
85.748.877.884.75814
399.258.4317
85.718.544.581.75814
388.157.9137
66.551.544.515.56614
373.563.9187
12.701.511.510.72114
363.281.4487
10.767.900.976.70114
35Prime!    2.999.9997
8.999.993.999.99813
341.484.8457
2.204.764.674.02213
331.000.0027
1.000.004.000.00113
32889.1906
790.658.856.09712
31299.9996
89.999.399.99811
30Prime!    249.3776
62.188.888.12611
29150.5256
22.657.775.62211
28148.5056
22.053.735.02211
27100.0026
10.000.400.00111
2689.3705
7.986.996.89710
2579.3275
6.292.772.92610
2442.8725
1.838.008.38110
2340.9585
1.677.557.76110
2240.4425
1.635.555.36110
2129.9995
899.939.9989
20Prime!    24.5475
602.555.2069
1912.9585
167.909.7619
1811.8625
140.707.0419
1710.0025
100.040.0019
16Prime!    2.9994
8.993.9987
151.8844
3.549.4537
141.0024
1.004.0017
132993
89.3985
121023
10.4015
11Prime!    832
6.8864
10382
1.4414
9Prime!    292
8383
8152
2223
7122
1413
661
332
5Prime!    51
222
4Prime!    31
61
3Prime!    21
11
211
–21
101
–31
up down
Case X = 4
See in-depth webpage about Palindromic Quasipronic Numbers of the form n(n+4)
One can find the regular numbers of the form n2 – 4 at
%N n2 – 4. under A028347.
The palindromic numbers of the form n2 – 4 are categorised as follows :
%N n2 – 4 is a palindrome. under A028555.
%N Palindromes of the form n2 – 4. under A028556.
211
–31
101
–41
up down
Case X = 5
One can find the regular numbers of the form n2 – 5 at
%N n2 – 5. under A028875.
The palindromic numbers of the form n2 – 5 are categorised as follows :
%N n2 – 5 is a palindrome. under A0?????.
%N Palindromes of the form n2 – 5. under A0?????.
70682.169.346.558.220.91318
465.355.017.383.670.106.601.076.383.710.553.56436
6997.596.148.796.869.70817
9.525.008.259.980.732.552.370.899.528.005.25934
6825.141.809.369.422.57917
632.110.578.368.384.979.483.863.875.011.23633
6711.794.718.752.417.45617
139.115.390.448.627.989.726.844.093.511.93133
663.147.974.405.060.84816
9.909.742.854.918.199.918.194.582.479.09931
653.060.109.141.231.76216
9.364.267.956.250.191.910.526.597.624.63931
642.480.391.882.989.53916
6.152.343.893.200.390.930.023.983.432.51631
631.177.345.302.316.09416
1.386.141.960.885.774.775.880.691.416.83131
62835.457.574.010.09915
697.989.357.970.840.048.079.753.989.79630
61370.527.710.594.14415
137.290.784.318.137.731.813.487.092.73130
60367.286.418.313.20615
134.899.313.077.343.343.770.313.998.43130
59311.724.544.575.57215
97.172.191.690.847.774.809.619.127.17929
58202.050.838.683.80315
40.824.541.412.828.182.821.414.542.80429
5797.888.964.408.30814
9.582.249.352.930.990.392.539.422.85928
56Prime!    79.518.245.402.17114
6.323.151.351.839.889.381.531.513.23628
5524.376.775.668.35014
594.227.191.985.060.589.191.722.49527
5420.991.089.109.00714
440.625.821.982.272.289.128.526.04427
5312.583.404.092.31614
158.342.058.550.515.055.850.243.85127
526.851.026.560.06313
46.936.564.926.688.662.946.563.96426
513.055.742.940.31213
9.337.564.917.266.627.194.657.33925
502.108.821.814.64313
4.447.129.445.914.195.449.217.44425
49682.055.723.83712
465.200.010.418.814.010.002.56424
48315.458.954.59812
99.514.352.036.063.025.341.59923
47309.549.710.80812
95.821.023.461.316.432.012.85923
46303.103.448.76812
91.871.700.655.055.600.717.81923
45251.972.689.52112
63.490.236.264.446.263.209.43623
44209.910.890.99312
44.062.582.157.475.128.526.04423
43140.638.946.71412
19.779.313.332.823.331.397.79123
42131.073.116.27612
17.180.161.810.301.816.108.17123
4198.841.904.32211
9.769.722.049.999.402.279.67922
4079.408.330.07111
6.305.682.884.664.882.865.03622
3931.283.616.87811
978.664.684.969.486.466.87921
3830.629.582.06211
938.171.297.292.792.171.83921
3724.849.903.93911
617.517.725.777.527.715.71621
3624.468.311.87011
598.698.285.767.582.896.89521
3520.990.990.99311
440.621.702.868.207.126.04421
3412.252.321.91611
150.119.392.333.293.911.05121
337.820.800.73910
61.164.924.199.142.946.11620
323.092.785.89210
9.565.324.573.754.235.65919
312.435.714.18010
5.932.703.566.653.072.39519
302.099.109.00710
4.406.258.623.268.526.04419
291.397.314.18610
1.952.486.934.396.842.59119
281.053.649.04610
1.110.176.312.136.710.11119
27210.900.8939
44.479.186.668.197.44417
2664.942.1238
4.217.479.339.747.12416
2521.089.1078
444.750.434.057.44415
2420.990.9938
440.621.787.126.04415
232.628.8517
6.910.857.580.19613
222.436.4207
5.936.142.416.39513
21978.5426
957.544.445.75912
20Prime!    783.3616
613.654.456.31612
19252.3796
63.695.159.63611
18210.8936
44.475.857.44411
1777.3705
5.986.116.89510
1669.4835
4.827.887.28410
15Prime!    21.6135
467.121.7649
1410.6965
114.404.4119
133.6564
13.366.3318
122.4404
5.953.5957
111.1564
1.336.3317
104363
190.0916
92593
67.0765
8812
6.5564
7142
Prime!    1913
6Prime!    71
442
541
Prime!    112
4Prime!    31
41
3Prime!    21
–11
211
–41
101
–51
up down
Case X = 6
One can find the regular numbers of the form n2 – 6 at
%N n2 – 6. under A028878.
The palindromic numbers of the form n2 – 6 are categorised as follows :
%N n2 – 6 is a palindrome. under A0?????.
%N Palindromes of the form n2 – 6. under A0?????.
69223.356.252.482.559.17018
49.888.015.523.052.719.391.725.032.551.088.89435
68190.010.989.109.890.01318
36.104.175.982.518.741.114.781.528.957.140.16335
67190.001.098.901.099.98718
36.100.417.583.625.578.687.552.638.571.400.16335
66190.000.109.989.000.01318
36.100.041.795.832.102.520.123.859.714.000.16335
65Prime!    190.000.010.890.000.01318
36.100.004.138.200.123.532.100.283.140.000.16335
64190.000.001.099.999.98718
36.100.000.417.999.996.269.999.971.400.000.16335
63190.000.000.000.000.01318
36.100.000.000.000.004.940.000.000.000.000.16335
6230.265.582.368.509.97517
916.005.476.105.061.868.160.501.674.500.61933
6123.029.141.616.879.67917
530.341.363.610.299.595.992.016.363.143.03533
6019.000.010.989.000.01317
361.000.417.582.121.252.121.285.714.000.16333
5919.000.000.000.000.01317
361.000.000.000.000.494.000.000.000.000.16333
581.900.108.910.890.01316
3.610.413.873.243.631.363.423.783.140.16331
571.900.010.998.900.01316
3.610.041.795.941.025.201.495.971.400.16331
561.900.010.891.099.98716
3.610.041.386.298.566.658.926.831.400.16331
551.900.001.089.000.01316
3.610.004.138.201.235.321.028.314.000.16331
541.900.000.109.999.98716
3.610.000.417.999.962.699.997.140.000.16331
53Prime!    1.900.000.000.000.01316
3.610.000.000.000.049.400.000.000.000.16331
52302.142.579.927.31515
91.290.138.605.133.933.150.683.109.21929
51296.255.835.953.97215
87.767.520.336.786.768.763.302.576.77829
50190.001.098.900.01315
36.100.417.583.212.521.238.571.400.16329
49190.000.000.000.01315
36.100.000.000.004.940.000.000.000.16329
4830.274.870.968.87514
916.567.812.182.030.281.218.765.61927
4730.166.868.936.75514
910.039.981.447.353.744.189.930.01927
4619.001.089.109.98714
361.041.387.365.666.563.783.140.16327
45Prime!    19.000.108.900.01314
361.004.138.212.353.212.831.400.16327
4419.000.010.999.98714
361.000.417.999.626.999.714.000.16327
4319.000.000.000.01314
361.000.000.000.494.000.000.000.16327
421.900.109.890.01313
3.610.417.594.125.214.957.140.16325
411.900.000.000.01313
3.610.000.000.049.400.000.000.16325
40227.544.887.93912
51.776.676.027.172.067.667.71523
39226.152.547.98912
51.144.974.961.916.947.944.11523
38221.502.874.81012
49.063.523.549.094.532.536.09423
37190.108.910.98712
36.141.398.036.663.089.314.16323
36190.010.890.01312
36.104.138.323.532.383.140.16323
35190.001.099.98712
36.100.417.996.269.971.400.16323
34190.000.000.01312
36.100.000.004.940.000.000.16323
33181.584.343.07712
32.972.873.650.705.637.827.92323
3230.181.812.65511
910.941.815.141.518.149.01921
3129.634.602.12211
878.209.642.929.246.902.87821
3024.452.608.65111
597.930.069.838.960.039.79521
2919.000.000.01311
361.000.000.494.000.000.16321
282.988.003.95210
8.928.167.617.167.618.29819
27Prime!    1.901.089.01310
3.614.139.435.349.314.16319
26Prime!    1.900.109.98710
Prime!    3.610.417.962.697.140.16319
251.900.000.01310
3.610.000.049.400.000.16319
24Prime!    191.098.9139
36.518.794.549.781.56317
23190.000.0139
36.100.004.940.000.16317
2229.848.3488
890.923.878.329.09815
2122.213.8808
493.456.464.654.39415
2019.010.9878
361.417.626.714.16315
19Prime!    19.000.0138
361.000.494.000.16315
182.930.6927
8.588.955.598.85813
172.316.1297
5.364.453.544.63513
162.220.5207
4.930.709.070.39413
15Prime!    1.900.7637
3.612.899.982.16313
141.900.0137
3.610.049.400.16313
13302.1356
91.285.558.21911
12191.0876
36.514.241.56311
11190.0136
Prime!    36.104.940.16311
1030.2755
916.575.6199
9Prime!    19.0135
361.494.1639
82.2614
5.112.1157
7Prime!    1.9134
3.659.5637
61.8574
3.448.4437
52923
85.2585
4Prime!    31
Prime!    31
3Prime!    21
–21
211
–51
101
–61
up down
Case X = 7
One can find the regular numbers of the form n2 – 7 at
%N n2 – 7. under A028881.
The palindromic numbers of the form n2 – 7 are categorised as follows :
%N n2 – 7 is a palindrome. under A0?????.
%N Palindromes of the form n2 – 7. under A0?????.
96314.309.382.828.649.28618
98.790.388.134.126.414.641.462.143.188.309.78935
95310.611.630.541.308.42618
Prime!    96.479.585.027.530.285.358.203.572.058.597.46935
94284.699.058.477.202.74518
81.053.553.897.805.708.180.750.879.835.535.01835
93279.000.109.989.000.12218
77.841.061.373.874.165.656.147.837.316.014.87735
92279.000.001.099.999.87818
77.841.000.613.799.933.133.999.731.600.014.87735
91213.060.011.257.408.23118
45.394.568.397.006.922.122.960.079.386.549.35435
90200.000.000.000.000.00018
39.999.999.999.999.999.999.999.999.999.999.99335
89199.976.057.913.842.10018
39.990.423.738.760.329.592.306.783.732.409.99335
88142.415.321.989.339.70318
20.282.123.937.327.304.740.372.373.932.128.20235
8727.900.010.989.000.12217
778.410.613.186.327.565.723.681.316.014.87733
8620.000.000.000.000.00017
399.999.999.999.999.999.999.999.999.999.99333
853.133.131.413.639.51416
9.816.512.455.134.739.374.315.542.156.18931
842.790.010.998.900.12216
7.784.161.373.983.656.563.893.731.614.87731
832.790.000.109.999.87816
7.784.100.613.799.331.339.973.160.014.87731
822.000.000.000.000.00016
3.999.999.999.999.999.999.999.999.999.99331
811.997.009.863.505.33016
3.988.048.394.937.576.757.394.938.408.89331
80307.703.972.678.78415
94.681.734.802.305.850.320.843.718.64929
79303.842.998.768.34415
92.320.567.900.539.893.500.976.502.32929
78279.001.098.900.12215
77.841.613.187.475.657.478.131.614.87729
77217.750.934.842.30915
47.415.469.624.699.499.642.696.451.47429
76208.939.304.470.62915
43.655.632.952.670.207.625.923.655.63429
75208.417.441.681.52115
43.437.829.997.070.207.079.992.873.43429
74200.000.000.000.00015
39.999.999.999.999.999.999.999.999.99329
7327.900.010.999.87814
778.410.613.793.313.397.316.014.87727
7222.217.531.211.79914
493.618.693.147.262.741.396.816.39427
7120.000.000.000.00014
399.999.999.999.999.999.999.999.99327
7015.689.702.164.40714
246.166.754.007.797.700.457.661.64227
693.010.534.949.20413
9.063.320.680.378.730.860.233.60925
682.847.385.777.70513
8.107.605.767.076.707.675.067.01825
672.790.109.890.12213
7.784.713.198.956.598.913.174.87725
662.000.000.000.00013
3.999.999.999.999.999.999.999.99325
651.674.399.042.96713
2.803.612.155.088.805.512.163.08225
641.640.692.403.06313
2.691.871.561.468.641.651.781.96225
631.438.749.187.14713
2.069.999.223.516.153.229.999.60225
621.416.714.309.24713
2.007.079.434.025.204.349.707.00225
61285.129.510.43512
81.298.837.720.902.773.889.21823
60279.001.099.87812
77.841.613.733.133.731.614.87723
59200.000.000.00012
39.999.999.999.999.999.999.99323
58198.455.185.22012
39.384.460.540.704.506.448.39323
57197.644.202.69012
39.063.230.856.965.803.236.09323
56169.804.109.16712
28.833.435.489.998.453.433.88223
5531.372.742.13611
984.248.949.131.949.842.48921
5420.000.000.00011
399.999.999.999.999.999.99321
5316.870.518.06711
284.614.379.848.973.416.48221
5214.895.699.37711
221.881.859.929.958.188.12221
512.790.109.87810
7.784.713.131.313.174.87719
50Prime!    2.088.111.87110
4.360.211.185.811.120.63419
492.048.625.66110
4.196.867.098.907.686.91419
482.000.000.00010
3.999.999.999.999.999.99319
471.508.870.97310
2.276.691.613.161.966.72219
461.417.005.75310
2.007.905.304.035.097.00219
45271.226.1129
Prime!    73.563.603.830.636.53717
44222.706.4499
49.598.162.426.189.59417
43200.000.0009
39.999.999.999.999.99317
42163.636.3639
26.776.859.295.867.76217
41161.682.7639
26.141.315.851.314.16217
40148.953.1279
22.187.034.043.078.12217
3928.495.9858
812.021.161.120.21815
3827.791.2228
772.352.020.253.27715
3720.000.0008
399.999.999.999.99315
362.850.1157
8.123.155.513.21813
352.787.9727
7.772.787.872.77713
342.780.2227
7.729.634.369.27713
332.143.7817
4.595.796.975.95413
322.000.0007
3.999.999.999.99313
311.996.6307
3.986.531.356.89313
30Prime!    1.636.3637
2.677.683.867.76213
291.569.8437
2.464.407.044.64213
28Prime!    212.9816
45.360.906.35411
27200.0006
39.999.999.99311
2627.1125
735.060.5379
2526.6825
Prime!    711.929.1179
2420.0005
399.999.9939
23Prime!    16.3635
267.747.7629
2215.5575
242.020.2429
212.0794
4.322.2347
20Prime!    2.0394
4.157.5147
192.0004
3.999.9937
18Prime!    1.6374
2.679.7627
171.5574
2.424.2427
163143
98.5895
152853
81.2185
142783
77.2775
132683
71.8175
122003
39.9935
11Prime!    1633
26.5625
10Prime!    1573
24.6425
9282
7773
8212
4343
7202
3933
6Prime!    172
2823
541
91
4Prime!    31
Prime!    21
3Prime!    21
–31
211
–61
101
–71
up down
Case X = 8
One can find the regular numbers of the form n2 – 8 at
%N n2 – 8. under A028884.
The palindromic numbers of the form n2 – 8 are categorised as follows :
%N n2 – 8 is a palindrome. under A0?????.
%N Palindromes of the form n2 – 8. under A0?????.
85255.601.656.911.517.44218
65.332.207.015.913.072.127.031.951.070.223.35635
84252.907.003.473.721.78818
63.961.952.406.057.124.542.175.060.425.916.93635
83199.999.999.999.999.99918
39.999.999.999.999.999.599.999.999.999.999.99335
82194.078.902.903.092.45918
37.666.620.552.067.990.509.976.025.502.666.67335
81183.337.479.874.002.37118
33.612.631.526.550.224.442.205.562.513.621.63335
80Prime!    100.000.000.000.000.00318
10.000.000.000.000.000.600.000.000.000.000.00135
7926.759.770.846.386.82517
716.085.335.751.134.252.431.157.533.580.61733
7819.999.999.999.999.99917
Prime!    399.999.999.999.999.959.999.999.999.999.99333
7710.313.548.756.006.85317
106.369.287.942.530.505.035.249.782.963.60133
7610.000.000.000.000.00317
100.000.000.000.000.060.000.000.000.000.00133
752.906.185.815.788.61616
8.445.915.995.890.943.490.985.995.195.44831
742.833.975.511.360.14616
8.031.417.198.989.001.009.898.917.141.30831
732.670.163.556.442.56516
7.129.773.418.154.007.004.518.143.779.21731
722.627.100.268.765.34816
6.901.655.822.146.963.696.412.285.561.09631
711.999.999.999.999.99916
3.999.999.999.999.995.999.999.999.999.99331
701.295.684.114.459.86316
1.678.797.324.463.639.363.644.237.978.76131
691.101.703.259.329.37316
1.213.750.071.616.963.696.161.700.573.12131
681.000.000.000.000.00316
1.000.000.000.000.006.000.000.000.000.00131
67199.999.999.999.99915
39.999.999.999.999.599.999.999.999.99329
66193.511.242.945.80915
37.446.601.146.431.913.464.110.664.47329
65181.886.803.762.17915
33.082.809.382.821.412.828.390.828.03329
64170.530.193.287.41015
29.080.546.822.641.414.622.864.508.09229
63130.122.803.816.31315
16.931.944.073.018.681.037.044.913.96129
62129.566.082.859.86315
16.787.369.827.648.884.672.896.378.76129
61100.000.000.000.00315
10.000.000.000.000.600.000.000.000.00129
6025.646.720.121.54214
657.754.252.992.707.299.252.457.75627
5919.999.999.999.99914
399.999.999.999.959.999.999.999.99327
5819.795.435.332.14914
391.859.259.989.292.989.952.958.19327
5718.339.849.698.37114
336.350.086.958.838.859.680.053.63327
5610.000.000.000.00314
100.000.000.000.060.000.000.000.00127
552.831.774.899.45413
8.018.949.081.177.711.809.498.10825
542.676.770.480.12513
7.165.100.203.268.623.020.015.61725
532.675.894.320.37513
7.160.410.413.815.183.140.140.61725
521.999.999.999.99913
3.999.999.999.995.999.999.999.99325
511.839.867.157.47113
Prime!    3.385.111.157.140.417.511.115.83325
501.010.273.888.29713
1.020.653.329.374.739.233.560.20125
491.000.000.000.00313
1.000.000.000.006.000.000.000.00125
48293.795.842.87612
86.315.997.291.219.279.951.36823
47266.623.495.19512
71.088.088.189.998.188.088.01723
46199.999.999.99912
39.999.999.999.599.999.999.99323
45173.192.892.40012
29.995.777.977.877.977.759.99223
44Prime!    100.000.000.00312
10.000.000.000.600.000.000.00123
4319.999.999.99911
399.999.999.959.999.999.99321
4219.295.962.59111
372.334.172.313.271.433.27321
4110.000.000.00311
100.000.000.060.000.000.00121
401.999.999.99910
3.999.999.995.999.999.99319
391.000.000.00310
1.000.000.006.000.000.00119
38286.740.1069
82.219.888.388.891.22817
37264.570.0029
69.997.285.958.279.99617
36199.999.9999
39.999.999.599.999.99317
35173.201.7009
29.998.828.882.889.99217
34127.813.0379
16.336.172.427.163.36117
33100.000.0039
10.000.000.600.000.00117
3229.731.0868
883.937.474.739.38815
31Prime!    19.999.9998
Prime!    399.999.959.999.99315
3010.000.0038
100.000.060.000.00115
292.871.2947
8.244.329.234.42813
281.999.9997
3.999.995.999.99313
27Prime!    1.211.5937
1.467.957.597.64113
261.184.2577
1.402.464.642.04113
251.121.9777
1.258.832.388.52113
24Prime!    1.000.0037
1.000.006.000.00113
23286.5566
82.114.341.12811
22Prime!    199.9996
39.999.599.99311
21120.8436
Prime!    14.603.030.64111
20Prime!    100.0036
10.000.600.00111
1928.3545
803.949.3089
1819.9995
399.959.9939
17Prime!    19.2595
370.909.0739
1617.2705
298.252.8929
1510.0035
Prime!    100.060.0019
142.5124
6.310.1367
13Prime!    1.9994
3.995.9937
12Prime!    1.3674
1.868.6817
11Prime!    1.1934
1.423.2417
101.0034
1.006.0017
9Prime!    1993
39.5935
8Prime!    1033
Prime!    10.6015
7Prime!    192
Prime!    3533
6Prime!    132
1613
541
81
4Prime!    31
11
3Prime!    21
–41
211
–71
101
–81
up down
Case X = 9
See in-depth webpage about Palindromic Quasipronic Numbers of the form n(n+6)
One can find the regular numbers of the form n2 – 9 at
%N n2 – 9. under A028560.
The palindromic numbers of the form n2 – 9 are categorised as follows :
%N n2 – 9 is a palindrome. under A028561.
%N Palindromes of the form n2 – 9. under A028562.
3Prime!    22
–53
212
–83
101
–91




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E-mail address : pdg@worldofnumbers.com