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Palindromic Nonagonals
(or 9-gonals) (or enneagonal)
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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

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

base x ( 7 x base - 5 )
-------------------------------
2


     PLAIN TEXT POLYGONS 


Normal and Palindromic Nonagonals

flash So far this compilation counts 68 Palindromic Nonagonals.
Here is the largest Sporadic Palindromic Nonagonal that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ September 27, 2024 ]

This basenumber
5.322.627.062.113.007.506.846.668.678
has 28 digits
yielding the following palindromic nonagonal number
99.156.255.948.182.109.153.103.300.355.300.330.135.190.128.184.955.265.199
with a length of 56 digits.

bu17 A palindromic nonagonal numbers can only end with digit 0, 1, 4, 5, 6, and 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
4 can be followed by any number : 40, 41, 42, 43, 44, 45, 46, 47, 48 or 49
5 can only be followed by 2 or 7 : 52 or 57
6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99

bu17 There exist no palindromic nonagonals of length
2, 6, 13, 14, 15, 16, 20, 25, 27, 28, 29, 30, 31, 32, 35, 37, 41, 44, 46, 48, 49, 51, 52, 53, 57.
(Sloane's A082722)

Case NonaChange of variablesCUDApalin parametersBase Correction
odd basen = 2 * m + 1
A B C →  14 9 1
CUDAbase * 2 + 1
even basen = 2 * m + 2
A B C →  14 23 9
CUDAbase * 2 + 2

bu17 Sloane's A048909 gives the first numbers that are both Nonagonal and Triangular.
1, 325, 82621, 20985481, 5330229625, 1353857339341, ...
Consult also Eric Weisstein's page Nonagonal Triangular Number.

bu17 Sloane's A036411 gives the first numbers that are both Nonagonal and Square.
1, 9, 1089, 8281, 978121, 7436529...
Consult also Eric Weisstein's page Nonagonal Square Number.

bu17 Sloane's A048915 gives the first numbers that are both Nonagonal and Pentagonal.
1, 651, 180868051, 95317119801, 26472137730696901, ...
Consult also Eric Weisstein's page Nonagonal Pentagonal Number.

bu17 Sloane's A048918 gives the first numbers that are both Nonagonal and Hexagonal.
1, 325, 5330229625, 1353857339341, 22184715227362706161, ...
Consult also Eric Weisstein's page Nonagonal Hexagonal Number.

bu17 Sloane's A048921 gives the first numbers that are both Nonagonal and Heptagonal.
1, 26884, 542041975, 10928650279834, ...
Consult also Eric Weisstein's page Nonagonal Heptagonal Number.

bu17 Sloane's A048924 gives the first numbers that are both Nonagonal and Octagonal.
1, 631125, 286703855361, 130242107189808901, ...
Consult also Eric Weisstein's page Nonagonal Octagonal Number.

bu17 The best way to get a 'structural' insight as how to imagine nonagonals is to visit for instance this site :


Sources Revealed


Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
One can find the regular nonagonal numbers at
%N 9-gonal (or enneagonal or nonagonal) numbers: n(7n–5)/2 under A001106.
The palindromic nonagonal numbers are categorised as follows :
%N n(7n–5)/2 is a palindromic nonagonal number under A055560.
%N Palindromic nonagonal numbers under A082723.
Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.


The Table


Index NrInfoBasenumberLength
Palindromic NonagonalsLength
   
[PG9] Formula = n(7n–5)/2
68Info5.322.627.062.113.007.506.846.668.67828
99.156.255.948.182.109.153.103.300.355.300.330.135.190.128.184.955.265.19956
67Info5.224.357.235.987.816.105.862.153.70228
95.528.679.852.263.887.827.488.512.533.521.588.472.878.836.225.897.682.55956
66Info1.876.119.816.345.331.309.680.746.92928
12.319.389.478.492.738.888.658.361.622.616.385.688.883.729.487.498.391.32156
65Info1.679.181.506.340.335.801.791.780.99328
9.868.776.859.323.897.215.794.481.742.471.844.975.127.983.239.586.778.68955
64Info1.362.471.141.388.162.023.922.065.80428
6.497.146.638.904.463.473.409.108.182.818.019.043.743.644.098.366.417.94655
63Info1.111.577.857.223.148.207.649.572.65228
4.324.618.664.340.819.816.841.803.754.573.081.486.189.180.434.668.164.23455
62Info371.144.372.961.223.228.658.331.22727
482.118.509.532.728.488.099.286.553.355.682.990.884.827.235.905.811.28454
61Info352.831.654.128.834.221.222.546.85227
435.715.616.543.512.545.618.365.652.256.563.816.545.215.345.616.517.53454
60Info2.056.602.438.516.214.335.318.94625
14.803.647.565.387.937.078.765.122.156.787.073.978.356.574.630.84150
59Info115.254.230.534.871.709.541.07224
46.492.381.796.648.740.224.886.168.842.204.784.669.718.329.46447
58Info12.771.589.611.884.159.330.40523
570.897.254.250.355.800.456.888.654.008.553.052.452.798.07545
57Info1.660.812.452.251.980.426.67322
9.654.043.005.443.328.676.169.616.768.233.445.003.404.56943
56Info1.639.652.478.336.096.510.56222
9.409.610.873.997.962.029.122.219.202.697.993.780.169.04943
55Info522.755.911.437.759.066.61321
956.458.100.300.927.701.829.928.107.729.003.001.854.65942
54Info198.343.896.557.195.374.94921
137.691.054.555.219.967.590.095.769.912.555.450.196.73142
53Info51.828.177.547.322.738.56220
9.401.559.957.568.830.940.550.490.388.657.599.551.04940
52Info21.052.483.436.370.199.81420
1.551.224.705.935.245.659.449.565.425.395.074.221.55140
51Info11.547.031.421.292.227.74720
466.668.771.255.085.018.151.810.580.552.177.866.66439
50Info10.973.370.993.981.638.98720
421.452.048.400.451.542.646.245.154.004.840.254.12439
49Info4.152.074.121.197.238.59919
60.339.018.277.705.374.211.247.350.777.281.093.30638
48Info3.672.784.064.195.630.08719
47.212.699.737.732.795.788.759.723.773.799.621.27438
47Info1.172.098.775.019.783.16319
4.808.354.384.410.066.932.396.600.144.834.538.08437
46Info587.730.406.426.154.76118
1.208.994.607.232.485.700.075.842.327.064.998.02137
45Info446.339.713.063.722.37918
697.266.988.102.321.088.880.123.201.889.662.79636
44Info18.287.529.802.802.80617
1.170.518.112.009.402.882.049.002.118.150.71134
43Info16.084.074.672.143.00217
905.441.103.206.752.020.257.602.301.144.50933
42Info14.107.415.463.308.17917
696.567.098.690.353.494.353.096.890.765.69633
41Info13.543.695.809.405.31117
642.010.936.621.960.404.069.126.639.010.24633
40Info13.267.696.935.194.77117
616.111.236.874.618.484.816.478.632.111.61633
39Info4.340.322.549.81613
65.934.399.427.533.572.499.343.95626
38Info3.696.823.830.48713
47.832.772.517.788.771.527.723.87426
37Info2.053.636.735.14613
14.760.983.439.788.793.438.906.74126
36Info386.688.531.61012
523.348.071.674.476.170.843.32524
35Info172.428.811.82612
104.060.933.016.610.339.060.40124
34Info139.168.803.13912
67.787.845.184.648.154.878.77623
33Info115.589.719.27212
46.763.441.204.540.214.436.76423
32Info75.356.090.49411
19.874.891.310.701.319.847.89123
31Info21.633.254.88111
1.637.992.008.558.002.997.36122
30Info11.947.307.36711
499.583.536.595.635.385.99421
29Info1.641.773.57810
9.433.971.680.861.793.34919
28Info1.121.857.19210
4.404.972.454.542.794.04419
27Info708.609.4299
1.757.445.628.265.447.57119
26Info521.037.0779
950.178.723.327.871.05918
25Info350.096.7879
428.987.160.061.789.82418
24Info168.566.8539
99.451.743.334.715.49917
23Info446.9046
699.030.030.99612
22Info435.6446
664.248.842.46612
21InfoPrime!    132.6196
61.556.965.51611
20Info52.6425
9.698.998.96910
19Info18.7295
1.227.667.22110
18Info16.8535
994.040.4999
17Info16.3185
931.929.1399
16Info12.8705
579.696.9759
15Info11.4725
460.595.0649
14Info11.3235
448.707.8449
13Info4.0704
57.966.9758
12Info2.1014
15.444.4518
11Info1.4114
6.964.6967
10InfoPrime!    1.1234
4.411.1447
9InfoPrime!    1393
67.2765
8Info742
18.9815
7Info442
6.6664
6InfoPrime!     172
9693
5Info122
4743
4Info61
1113
3InfoPrime!     21
91
2Info11
11
1Info01
01


Contributions

Feng Yuan (email) from Washington State, USA, discovered the palindromic nonagonals
starting from index number [1] up to [44].

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, discovered the palindromic nonagonals
starting from index number [45] up to [52].

Higher values up to length 47 were found and submitted by Feng Yuan.

Exhaustive (re)-search performed up to length 36 by Patrick De Geest and found no missers.

[ November 4, 2022 ]
Robert Xiao (email) added four missing entries [46], [47], [49] and [53].

[ November 11, 2022 ]
Robert Xiao (email) added two missing entries [56] and [57].

[ November 13, 2022 ]
David Griffeath (email) using Rust code by Robert Xiao (email) discovered a record entry [60].

[ December 6, 2022 ]
Robert Xiao (email) added one missing entry [58].

[ January 11, 2023 ]
David Griffeath (email) using CUDA code by Robert Xiao (email) did a search for 53-digit palindromic nonagonals but there are none.
So he pushed through to 54 digits and found two new record entries [61] and [62], one with even root and one with odd root.

[ June 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [63] up to [65].

[ September 27, 2024 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [66] up to [68].

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( © All rights reserved ) - Last modified : September 27, 2024.
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E-mail address : pdg@worldofnumbers.com