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, 34086761467054596643044235

Offset

1,2

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 40.

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)).

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

Limit_{n->oo} a(2*n)/a(2*n-1) = (1/7)*(43 + 30*sqrt(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: A386556 A073903 A069710 * A234634 A004811 A227323

Adjacent sequences: A046184 A046185 A046186 * A046188 A046189 A046190

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved