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

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

Offset: 1

Author

Colin Barker, Jan 03 2015

Keywords

• nonn
• easy

Examples

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

Links

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

Crossrefs

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

Programs

PARI

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

Formula

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)