A046187 Indices of pentagonal numbers which are also octagonal.

1, 11, 1025, 12507, 1182657, 14432875, 1364784961, 16655525051, 1574960662145, 19220461475787, 1817503239330177, 22180395887532955, 2097397163226361921, 25596157633751554091, 2420394508859982326465, 29537943728953405887867, 2793133165827256378378497

Offset

1,2

Comments

From Ant King, Dec 16 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 16 2011: (Start)

a(n) = 1154*a(n-2) - a(n-4) - 192.

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

a(n) = (1/12)*((3-sqrt(2)*(-1)^n)*(1+sqrt(2))^(4*n-3) + (3 + sqrt(2)*(-1)^n)*(1-sqrt(2))^(4*n-3) + 2).

a(n) = ceiling((1/12)*(3-sqrt(2)*(-1)^n)*(1+sqrt(2))^(4*n-3)).

G.f.: x*(1-8*x-x^2)*(1+18*x+5*x^2)/((1-x)*(1-34*x+x^2)*(1+34*x+x^2)).

(End)

Mathematica

LinearRecurrence[{1, 1154, -1154, -1, 1}, {1, 11, 1025, 12507, 1182657}, 15] (* Ant King, Dec 16 2011 *)

Prog

(PARI) Vec(x*(x^2+8*x-1)*(5*x^2+18*x+1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^50)) \\ Colin Barker, Jun 23 2015

Crossrefs

Cf. A046188, A046189.

Sequence in context: A099440 A073903 A069710 * A234634 A004811 A227323

Adjacent sequences: A046184 A046185 A046186 * A046188 A046189 A046190

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved