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

     PLAIN TEXT POLYGONS 


Normal and Palindromic Nonagonals

flash So far I compiled 52 Palindromic Nonagonals.
These were all found by Feng Yuan from Washington State, USA.
Here is the largest one sent in on [ January 8, 2008 ]

This basenumber
115.254.230.534.871.709.541.072

has 24 digits
yielding the following palindromic nonagonal number
46.492.381.796.648.740.224.886.168.842.204.784.669.718.329.464
with a length of 47 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, 43, 44, 45, 46.
(Sloane's A082722)


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.


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

Exhaustive search performed upto length 33 by Patrick De Geest
Higher values up to length 47 were found and submitted by Feng Yuan.



Index NrInfo BasenumberLength
Palindromic NonagonalsLength
   
Formula = n(7n–5)/2
52 Info 115.254.230.534.871.709.541.07224
46.492.381.796.648.740.224.886.168.842.204.784.669.718.329.46447
51 Info522.755.911.437.759.066.61321
956.458.100.300.927.701.829.928.107.729.003.001.854.65942
50 Info198.343.896.557.195.374.94921
137.691.054.555.219.967.590.095.769.912.555.450.196.73142
49 Info21.052.483.436.370.199.81420
1.551.224.705.935.245.659.449.565.425.395.074.221.55140
48 Info11.547.031.421.292.227.74720
466.668.771.255.085.018.151.810.580.552.177.866.66439
47 Info10.973.370.993.981.638.98720
421.452.048.400.451.542.646.245.154.004.840.254.12439
46 Info3.672.784.064.195.630.08719
47.212.699.737.732.795.788.759.723.773.799.621.27438
45 Info446.339.713.063.722.37918
697.266.988.102.321.088.880.123.201.889.662.79636
44 Info18.287.529.802.802.80617
1.170.518.112.009.402.882.049.002.118.150.71134
43 Info16.084.074.672.143.00217
905.441.103.206.752.020.257.602.301.144.50933
42 Info14.107.415.463.308.17917
696.567.098.690.353.494.353.096.890.765.69633
41 Info13.543.695.809.405.31117
642.010.936.621.960.404.069.126.639.010.24633
40 Info13.267.696.935.194.77117
616.111.236.874.618.484.816.478.632.111.61633
39 Info4.340.322.549.81613
65.934.399.427.533.572.499.343.95626
38 Info3.696.823.830.48713
47.832.772.517.788.771.527.723.87426
37 Info2.053.636.735.14613
14.760.983.439.788.793.438.906.74126
36 Info386.688.531.61012
523.348.071.674.476.170.843.32524
35 Info172.428.811.82612
104.060.933.016.610.339.060.40124
34 Info139.168.803.13912
67.787.845.184.648.154.878.77623
33 Info115.589.719.27212
46.763.441.204.540.214.436.76423
32 Info75.356.090.49411
19.874.891.310.701.319.847.89123
31 Info21.633.254.88111
1.637.992.008.558.002.997.36122
30 Info11.947.307.36711
499.583.536.595.635.385.99421
29 Info1.641.773.57810
9.433.971.680.861.793.34919
28 Info1.121.857.19210
4.404.972.454.542.794.04419
27 Info708.609.4299
1.757.445.628.265.447.57119
26 Info521.037.0779
950.178.723.327.871.05918
25 Info350.096.7879
428.987.160.061.789.82418
24 Info168.566.8539
99.451.743.334.715.49917
23 Info446.9046
699.030.030.99612
22 Info435.6446
664.248.842.46612
21 InfoPrime!    132.6196
61.556.965.51611
20 Info52.6425
9.698.998.96910
19 Info18.7295
1.227.667.22110
18 Info16.8535
994.040.4999
17 Info16.3185
931.929.1399
16 Info12.8705
579.696.9759
15 Info11.4725
460.595.0649
14 Info11.3235
448.707.8449
13 Info4.0704
57.966.9758
12 Info2.1014
15.444.4518
11 Info1.4114
6.964.6967
10 InfoPrime!    1.1234
4.411.1447
9 InfoPrime!    1393
67.2765
8 Info742
18.9815
7 Info442
6.6664
6 InfoPrime!     172
9693
5 Info122
4743
4 Info61
1113
3 InfoPrime!     21
91
2 Info11
11
1 Info01
01


Contributions

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

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





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( © All rights reserved ) - Last modified : September 2, 2021.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com