A101321 -id:A101321 - cp's OEIS frontend

A339010 a(n) is the number of ways to write n as the difference of two centered k-gonal numbers for k >= 3.

0, 0, 1, 1, 1, 2, 1, 2, 3, 2, 1, 5, 1, 2, 5, 3, 1, 6, 1, 5, 5, 2, 1, 8, 3, 2, 6, 5, 1, 10, 1, 4, 5, 2, 5, 12, 1, 2, 5, 8, 1, 10, 1, 5, 12, 2, 1, 11, 3, 6, 5, 5, 1, 12, 5, 8, 5, 2, 1, 19, 1, 2, 12, 5, 5, 10, 1, 5, 5, 10, 1, 18, 1, 2, 12, 5, 5, 10, 1, 11, 10, 2

Offset: 1

Peter Kagey, Nov 18 2020

nonn

Records occur at indices n = 1, 3, 6, 9, 12, 18, 24, 30, 36, 60, 90, 120, 180, 270, 360, 420, 540, 630, 840, 1080, ...

			For n = 35, the a(35) = 5 differences are:
A101321( 5,4) - A101321( 5,2) =  51 -  16 = 35,
A101321( 5,7) - A101321( 5,6) = 141 - 106 = 35,
A101321( 7,3) - A101321( 7,1) =  43 -   8 = 35,
A101321( 7,5) - A101321( 7,4) = 106 -  71 = 35, and
A101321(36,1) - A101321(36,0) =  36 -   1 = 35.

Peter Kagey, Table of n, a(n) for n = 1..10000
Code Golf Stack Exchange, Uncentered Polygons
OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Wikipedia, Centered polygonal number

Cf. A001227, A101321.

Cf. A333822 (polygonal numbers), A333836 (positive polygonal numbers), A333868 (binomial coefficients), A333880 (perfect powers).

a(n) = sumdiv(n, d, if (3*d <= n, numdiv(d>>valuation(d, 2)))); \\ Michel Marcus, Nov 19 2020

a(n) = Sum_{d|n, 3*d <= n} A001227(d).

A352481 Number of positive integers that are not the sum of distinct centered n-gonal numbers.

Original entry on oeis.org

42, 84, 148, 237, 363, 534, 766, 1060, 1436, 1911, 2484, 3184, 4014

Offset: 3

Ilya Gutkovskiy, Mar 21 2022

Eric Weisstein's World of Mathematics, Centered Polygonal Number

Cf. A025524, A101321, A352349, A352585.

A345038 Triangle T(n,k) read by rows of the smallest centered n-gonal number greater than 1 that is also centered k-gonal, or 0 if none exists, for 1 <= k <= n.

Original entry on oeis.org

2, 7, 3, 4, 31, 4, 0, 13, 85, 5, 16, 31, 31, 181, 6, 7, 7, 19, 61, 331, 7, 22, 43, 316, 841, 106, 547, 8, 121, 0, 361, 25, 22801, 169, 841, 9, 0, 91, 10, 0, 1891, 91, 253, 1225, 10, 11, 31, 31, 61, 31, 61, 2101, 361, 1711, 11, 67, 111, 166, 8581, 1156, 397, 6931, 179479609, 496, 2311, 12

Offset: 1

Mohammed Yaseen, Jun 06 2021

The i-th centered j-gonal number is j*i*(i-1)/2 + 1. Thus if the p-th centered n-gonal number is also the q-th centered k-gonal number, then n*p*(p-1) = k*q*(q-1). Therefore T(n,k) = n*p*(p-1)/2 + 1 = k*q*(q-1)/2 + 1 iff n*p*(p-1) = k*q*(q-1) has a nontrivial positive integer solution. Otherwise T(n,k) = 0. It also implies that when T(n,k) = 0, T(r*n,r*k) = 0 for any positive integer r.

			The triangle begins:
   2;
   7,   3;
   4,  31,   4;
   0,  13,  85,   5;
  16,  31,  31, 181,   6;
   7,   7,  19,  61, 331,   7;
  22,  43, 316, 841, 106, 547,   8;
  ...

Index entries for sequences related to centered polygonal numbers

Cf. A101321, A189216.

iszero(n,k)={if(issquare(n) && issquare(k) && n<>k, my(t=n-k); fordiv(t, d, my(p=(d+t/d)/2/sqrtint(n), q=(d-t/d)/2/sqrtint(k)); if(abs(p)!=1 && !frac(p) && !frac(q) && p%2==1 && q%2==1, return(0))); 1, 0)}
T(n, k)={my(g=gcd(n,k)); n/=g; k/=g; if(iszero(n, k), 0, for(p=2, oo, my(t=n*p*(p-1)/2); if(t%k==0 && ispolygonal(t/k, 3), return(t*g+1))))} \\ Andrew Howroyd, Jun 08 2021

T(n,n) = n+1.

A352975 a(n) is the smallest number which can be represented as the sum of n distinct centered n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

96, 192, 330, 504, 840, 1304, 1872, 2910, 3971, 5340, 6851, 8932, 11700, 14496, 18258, 22410, 27265, 32620, 39606, 47124, 55545, 65448, 76050, 87854, 101925, 116956, 134125, 152340, 173538, 195424, 220473, 246942, 276570, 306756, 340918, 377644, 418821, 462720

Offset: 3

Ilya Gutkovskiy, Apr 13 2022

nonn

If a(n) exists, then n divides a(n). - Thomas Scheuerle, Apr 13 2022

			For n = 3: 96 = 1 + 10 + 85 = 1 + 31 + 64 = 19 + 31 + 46.

Eric Weisstein's World of Mathematics, Centered Polygonal Number

Cf. A101321, A350209, A350397, A350405.

a(n) >= n*binomial(n + 2, 3) + n, if a(n) exists. - Thomas Scheuerle, Apr 13 2022

a(10)-a(16) from Thomas Scheuerle, Apr 13 2022

a(17)-a(40) from Michael S. Branicky, May 19 2022

A360663 a(n) is the least integer m >= 3 such that n is a centered m-gonal number.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 3, 10, 11, 4, 13, 14, 5, 16, 17, 3, 19, 20, 7, 22, 23, 4, 25, 26, 9, 28, 29, 3, 31, 32, 11, 34, 35, 6, 37, 38, 13, 4, 41, 7, 43, 44, 3, 46, 47, 8, 49, 5, 17, 52, 53, 9, 55, 56, 19, 58, 59, 4, 61, 62, 3, 64, 65, 11, 67, 68, 23, 7, 71, 12, 73, 74, 5, 76, 77, 13, 79

Offset: 4

Ilya Gutkovskiy, Feb 15 2023

nonn

			a(16) = 5 since 16 is a centered pentagonal number, but not a centered square or centered triangular number.

Eric Weisstein's World of Mathematics, Centered Polygonal Number.
Wikipedia, Centered polygonal number.

Cf. A101321, A176774, A333914.

seq[len_] := Module[{s = Table[0, {len}], c = 0, k = 3, n, ckn}, While[c < len, n = 2; While[(ckn = k*n*(n - 1)/2 - 2) <= len, If[s[[ckn]] == 0, c++; s[[ckn]] = k]; n++]; n = 4; k++]; s]; seq[100] (* Amiram Eldar, Mar 06 2023 *)

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A339010 a(n) is the number of ways to write n as the difference of two centered k-gonal numbers for k >= 3.

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A352481 Number of positive integers that are not the sum of distinct centered n-gonal numbers.

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A345038 Triangle T(n,k) read by rows of the smallest centered n-gonal number greater than 1 that is also centered k-gonal, or 0 if none exists, for 1 <= k <= n.

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A352975 a(n) is the smallest number which can be represented as the sum of n distinct centered n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A360663 a(n) is the least integer m >= 3 such that n is a centered m-gonal number.

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