A048914 Indices of pentagonal numbers which are also 9-gonal.

1, 21, 10981, 252081, 132846121, 3049673901, 1607172358861, 36894954600201, 19443571064652241, 446355157703555781, 235228321132990450741, 5400004661002663236321, 2845792209623347408410361, 65329255942455062129453661, 34428393916794935813958094621

Offset

1,2

Comments

From Ant King, Dec 20 2011: (Start)

lim(n->Infinity, a(2n+1)/a(2n))=1/2*(527+115*sqrt(21))

lim(n->Infinity, a(2n)/a(2n-1))=1/2*(23+5*sqrt(21))

(End)

Links

Colin Barker, Table of n, a(n) for n = 1..490

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

Index entries for linear recurrences with constant coefficients, signature (1,12098,-12098,-1,1).

Formula

From Ant King, Dec 20 2011: (Start)

a(n) = 12098*a(n-2)-a(n-4)-2016.

a(n) = a(n-1)+12098*a(n-2)-12098*a(n-3)-a(n-4)+a(n-5).

a(n) = 1/84*((2+sqrt(21))*(7-sqrt(21)*(-1)^n)*(2*sqrt(7)+3*sqrt(3))^(2n-2)+ (2-sqrt(21))*(7+sqrt(21)*(-1)^n)*(2*sqrt(7)-3*sqrt(3))^(2n-2)+14).

a(n) = ceiling(1/84*(2+sqrt(21))*(7-sqrt(21)*(-1)^n)*(2*sqrt(7)+3*sqrt(3))^(2n-2)).

G.f.: x*(1+20*x-1138*x^2-860*x^3-39*x^4) / ((1-x)*(1-110*x+x^2)*(1+110*x+x^2))

(End)

Mathematica

LinearRecurrence[{1, 12098, -12098, -1, 1}, {1, 21, 10981, 252081, 132846121}, 13] (* Ant King, Dec 20 2011 *)

Prog

(PARI) Vec(x*(39*x^4+860*x^3+1138*x^2-20*x-1)/((x-1)*(x^2-110*x+1)*(x^2+110*x+1)) + O(x^20)) \\ Colin Barker, Jun 22 2015

Crossrefs

Cf. A048913, A048915.

Sequence in context: A135823 A013726 A159358 * A046183 A203674 A250065

Adjacent sequences: A048911 A048912 A048913 * A048915 A048916 A048917

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved