A253447 -id:A253447 - cp's OEIS frontend

A253446 Indices of centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

1, 16, 465, 13920, 417121, 12499696, 374573745, 11224712640, 336366805441, 10079779450576, 302057016711825, 9051630721904160, 271246864640412961, 8128354308490484656, 243579382390074126705, 7299253117393733316480, 218734014139421925367681

Offset: 1

Colin Barker, Jan 01 2015

Also positive integers x in the solutions to 7*x^2 - 8*y^2 - 7*x + 8*y = 0, the corresponding values of y being A253447.

			16 is in the sequence because the 16th centered heptagonal number is 841, which is also the 15th centered octagonal number.

Colin Barker, Table of n, a(n) for n = 1..678
Index entries for linear recurrences with constant coefficients, signature (31,-31,1).

Cf. A016754, A069099, A253447, A253514.

LinearRecurrence[{31,-31,1},{1,16,465},20] (* Harvey P. Dale, Oct 04 2023 *)

Vec(x*(15*x-1)/((x-1)*(x^2-30*x+1)) + O(x^100))

a(n) = 31*a(n-1)-31*a(n-2)+a(n-3).
G.f.: x*(15*x-1) / ((x-1)*(x^2-30*x+1)).
a(n) = sqrt((-2-(15-4*sqrt(14))^n-(15+4*sqrt(14))^n)*(2-(15-4*sqrt(14))^(1+n)-(15+4*sqrt(14))^(1+n)))/(4*sqrt(7)).  - Gerry Martens, Jun 04 2015

A253514 Centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 841, 755161, 678133681, 608963290321, 546848356574521, 491069215240629481, 440979608437728699361, 395999197307865131396641, 355606838202854450265484201, 319334544706965988473273415801, 286762065540017254794549261905041

Offset: 1

Colin Barker, Jan 03 2015

			841 is in the sequence because it is the 16th centered heptagonal number and the 15th centered octagonal number.

Colin Barker, Table of n, a(n) for n = 1..339
Index entries for linear recurrences with constant coefficients, signature (899,-899,1).

Cf. A001110, A016754, A069099, A157877, A157879, A253446, A253447.

Vec(-x*(x^2-58*x+1)/((x-1)*(x^2-898*x+1)) + O(x^100))

a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).
G.f.: -x*(x^2-58*x+1) / ((x-1)*(x^2-898*x+1)).
From Peter Bala, Apr 15 2025; (Start)
a(n) = (1/64)*(-4 + sqrt(14))^2*(15 + 4*sqrt(14) + (449 + 120*sqrt(14))^n)^2 *(449 + 120*sqrt(14))^(-n).
a(-n) = a(n+1).
a(n) = (1/16) * (1 - T(2*n+1, -15)), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. Cf. A001110.
a(n) = A157877(n)^2 = 1 + 7*A157879(n).
a(2) divides a(3*n+2); a(3) divides a(5*n+3); a(4) divides a(7*n+4); a(5) divides a(9*n+5). In general, a(k) divides a((2*k-1)*n + k). (End)

cp's OEIS Frontend

A253446 Indices of centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

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