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A046188
Indices of octagonal numbers which are also pentagonal.
3
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
Eric Weisstein's World of Mathematics, Octagonal Pentagonal Number.
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
Sequence in context: A277860 A221198 A071308 * A014387 A220641 A017007
KEYWORD
nonn,easy
STATUS
approved