A046188 Indices of octagonal numbers which are also pentagonal.

1, 8, 725, 8844, 836265, 10205584, 965048701, 11777234708, 1113665364305, 13590918647064, 1285168865358885, 15683908341476764, 1483083756958788601, 18099216635145538208, 1711477370361576686285, 20886480313049609614884, 1975043402313502537183905

Offset

1,2

Comments

From Ant King, Dec 17 2011: (Start)

lim_{n->infinity} a(2n+1)/a(2n) = (1/7)*(331 + 234*sqrt(2)).

lim_{n->infinity} a(2n)/a(2n-1) = (1/7)*(43 + 30*sqrt(2)).

(End)

Links

Colin Barker, Table of n, a(n) for n = 1..654

Eric Weisstein's World of Mathematics, Octagonal Pentagonal Number.

Index entries for linear recurrences with constant coefficients, signature (1,1154,-1154,-1,1).

Formula

From Ant King, Dec 17 2011: (Start)

a(n) = 1154*a(n-2) - a(n-4) - 384.

a(n) = a(n-1) + 1154*a(n-2) - 1154*a(n-3) - a(n-4) + a(n-5).

a(n) = (1/24)*sqrt(2)*((3-sqrt(2)*(-1)^n)*(1+sqrt(2))^(4*n-3) - (3+sqrt(2)*(-1)^n)*(1-sqrt(2))^(4*n-3) + 4*sqrt(2)).

a(n) = ceiling((1/24)*sqrt(2)*((3-sqrt(2)*(-1)^n)*(1+sqrt(2))^(4*n-3))).

G.f.: x*(1 + 7*x - 437*x^2 + 41*x^3 + 4*x^4)/((1-x)*(1 - 34*x + x^2)*(1 + 34*x + x^2)).

(End)

Mathematica

LinearRecurrence[{1, 1154, -1154, -1, 1}, {1, 8, 725, 8844, 836265}, 15] (* Ant King, Dec 17 2011 *)

Prog

(PARI) Vec(-x*(4*x^4+41*x^3-437*x^2+7*x+1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^50)) \\ Colin Barker, Jun 23 2015

Crossrefs

Cf. A046187, A046189.

Sequence in context: A277860 A221198 A071308 * A014387 A220641 A017007

Adjacent sequences: A046185 A046186 A046187 * A046189 A046190 A046191

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved