More about Palindromic Nonagonals

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Palindromic Nonagonals (or 9-gonals) (or enneagonal)
Factorization Records triangle square penta hexa hepta octa

Introduction

Palindromic numbers are numbers which read the same from

left to right (forwards) as from the right to left (backwards)

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 :

From Eric Weisstein's Math Encyclopedia : Nonagonal Number

Normal and Palindromic Nonagonals

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.

A palindromic nonagonal numbers can only end with digit 1, 4, 5, 6, and 9.

There exist no palindromic nonagonals of length

2, 6, 13, 14, 15, 16, 20, 25, 27, 28, 29, 30, 31, 32, ...

(Sloane's A082722)

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

Index Nr	Info	Basenumber	Length
Index Nr	Palindromic Nonagonals		Length

	Formula = n(7n–5)/2
51	Info	115.254.230.534.871.709.541.072	24
51	46.492.381.796.648.740.224.886.168.842.204.784.669.718.329.464		47
50	Info	522.755.911.437.759.066.613	21
50	956.458.100.300.927.701.829.928.107.729.003.001.854.659		42
49	Info	198.343.896.557.195.374.949	21
49	137.691.054.555.219.967.590.095.769.912.555.450.196.731		42
48	Info	21.052.483.436.370.199.814	20
48	1.551.224.705.935.245.659.449.565.425.395.074.221.551		40
47	Info	11.547.031.421.292.227.747	20
47	466.668.771.255.085.018.151.810.580.552.177.866.664		39
46	Info	10.973.370.993.981.638.987	20
46	421.452.048.400.451.542.646.245.154.004.840.254.124		39
45	Info	3.672.784.064.195.630.087	19
45	47.212.699.737.732.795.788.759.723.773.799.621.274		38
44	Info	446.339.713.063.722.379	18
44	697.266.988.102.321.088.880.123.201.889.662.796		36
43	Info	18.287.529.802.802.806	17
43	1.170.518.112.009.402.882.049.002.118.150.711		34
42	Info	16.084.074.672.143.002	17
42	905.441.103.206.752.020.257.602.301.144.509		33
41	Info	14.107.415.463.308.179	17
41	696.567.098.690.353.494.353.096.890.765.696		33
40	Info	13.543.695.809.405.311	17
40	642.010.936.621.960.404.069.126.639.010.246		33
39	Info	13.267.696.935.194.771	17
39	616.111.236.874.618.484.816.478.632.111.616		33
38	Info	4.340.322.549.816	13
38	65.934.399.427.533.572.499.343.956		26
37	Info	3.696.823.830.487	13
37	47.832.772.517.788.771.527.723.874		26
36	Info	2.053.636.735.146	13
36	14.760.983.439.788.793.438.906.741		26
35	Info	386.688.531.610	12
35	523.348.071.674.476.170.843.325		24
34	Info	172.428.811.826	12
34	104.060.933.016.610.339.060.401		24
33	Info	139.168.803.139	12
33	67.787.845.184.648.154.878.776		23
32	Info	115.589.719.272	12
32	46.763.441.204.540.214.436.764		23
31	Info	75.356.090.494	11
31	19.874.891.310.701.319.847.891		23
30	Info	21.633.254.881	11
30	1.637.992.008.558.002.997.361		22
29	Info	11.947.307.367	11
29	499.583.536.595.635.385.994		21
28	Info	1.641.773.578	10
28	9.433.971.680.861.793.349		19
27	Info	1.121.857.192	10
27	4.404.972.454.542.794.044		19
26	Info	708.609.429	9
26	1.757.445.628.265.447.571		19
25	Info	521.037.077	9
25	950.178.723.327.871.059		18
24	Info	350.096.787	9
24	428.987.160.061.789.824		18
23	Info	168.566.853	9
23	99.451.743.334.715.499		17
22	Info	446.904	6
22	699.030.030.996		12
21	Info	435.644	6
21	664.248.842.466		12
20	Info	Prime! 132.619	6
20	61.556.965.516		11
19	Info	52.642	5
19	9.698.998.969		10
18	Info	18.729	5
18	1.227.667.221		10
17	Info	16.853	5
17	994.040.499		9
16	Info	16.318	5
16	931.929.139		9
15	Info	12.870	5
15	579.696.975		9
14	Info	11.472	5
14	460.595.064		9
13	Info	11.323	5
13	448.707.844		9
12	Info	4.070	4
12	57.966.975		8
11	Info	2.101	4
11	15.444.451		8
10	Info	1.411	4
10	6.964.696		7
9	Info	Prime! 1.123	4
9	4.411.144		7
8	Info	Prime! 139	3
8	67.276		5
7	Info	74	2
7	18.981		5
6	Info	44	2
6	6.666		4
5	Info	Prime! 17	2
5	969		3
4	Info	12	2
4	474		3
3	Info	6	1
3	111		3
2	Info	Prime! 2	1
2	9		1
1	Info	1	1
1	1		1

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.

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.

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.

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.

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.

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.

Contributions

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

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

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