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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 :


Normal and Palindromic Nonagonals

flash So far I compiled 51 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 1, 4, 5, 6, and 9.

bu17 There exist no palindromic nonagonals of length
2, 6, 13, 14, 15, 16, 20, 25, 27, 28, 29, 30, 31, 32, ...
(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
51 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
50 Info522.755.911.437.759.066.61321
956.458.100.300.927.701.829.928.107.729.003.001.854.65942
49 Info198.343.896.557.195.374.94921
137.691.054.555.219.967.590.095.769.912.555.450.196.73142
48 Info21.052.483.436.370.199.81420
1.551.224.705.935.245.659.449.565.425.395.074.221.55140
47 Info11.547.031.421.292.227.74720
466.668.771.255.085.018.151.810.580.552.177.866.66439
46 Info10.973.370.993.981.638.98720
421.452.048.400.451.542.646.245.154.004.840.254.12439
45 Info3.672.784.064.195.630.08719
47.212.699.737.732.795.788.759.723.773.799.621.27438
44 Info446.339.713.063.722.37918
697.266.988.102.321.088.880.123.201.889.662.79636
43 Info18.287.529.802.802.80617
1.170.518.112.009.402.882.049.002.118.150.71134
42 Info16.084.074.672.143.00217
905.441.103.206.752.020.257.602.301.144.50933
41 Info14.107.415.463.308.17917
696.567.098.690.353.494.353.096.890.765.69633
40 Info13.543.695.809.405.31117
642.010.936.621.960.404.069.126.639.010.24633
39 Info13.267.696.935.194.77117
616.111.236.874.618.484.816.478.632.111.61633
38 Info4.340.322.549.81613
65.934.399.427.533.572.499.343.95626
37 Info3.696.823.830.48713
47.832.772.517.788.771.527.723.87426
36 Info2.053.636.735.14613
14.760.983.439.788.793.438.906.74126
35 Info386.688.531.61012
523.348.071.674.476.170.843.32524
34 Info172.428.811.82612
104.060.933.016.610.339.060.40124
33 Info139.168.803.13912
67.787.845.184.648.154.878.77623
32 Info115.589.719.27212
46.763.441.204.540.214.436.76423
31 Info75.356.090.49411
19.874.891.310.701.319.847.89123
30 Info21.633.254.88111
1.637.992.008.558.002.997.36122
29 Info11.947.307.36711
499.583.536.595.635.385.99421
28 Info1.641.773.57810
9.433.971.680.861.793.34919
27 Info1.121.857.19210
4.404.972.454.542.794.04419
26 Info708.609.4299
1.757.445.628.265.447.57119
25 Info521.037.0779
950.178.723.327.871.05918
24 Info350.096.7879
428.987.160.061.789.82418
23 Info168.566.8539
99.451.743.334.715.49917
22 Info446.9046
699.030.030.99612
21 Info435.6446
664.248.842.46612
20 InfoPrime!    132.6196
61.556.965.51611
19 Info52.6425
9.698.998.96910
18 Info18.7295
1.227.667.22110
17 Info16.8535
994.040.4999
16 Info16.3185
931.929.1399
15 Info12.8705
579.696.9759
14 Info11.4725
460.595.0649
13 Info11.3235
448.707.8449
12 Info4.0704
57.966.9758
11 Info2.1014
15.444.4518
10 Info1.4114
6.964.6967
9 InfoPrime!    1.1234
4.411.1447
8 InfoPrime!    1393
67.2765
7 Info742
18.9815
6 Info442
6.6664
5 InfoPrime!     172
9693
4 Info122
4743
3 Info61
1113
2 InfoPrime!     21
91
1 Info11
11


Contributions

Feng Yuan (email) from Washington State, USA, discovered the palindromic nonagonals
starting from index number 44 up to 51 [ January 8, 2008 ].

Feng Yuan (email) from Washington State, USA, discovered the palindromic nonagonals
starting from index number 1 up to 43.





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Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com