A048903 Heptagonal hexagonal numbers.

1, 121771, 12625478965, 1309034909945503, 135723357520344181225, 14072069153115290487843091, 1459020273797576190840203197981, 151274140013808225465578657485241095, 15684405383452644158924550174544564031953, 1626190518815862911671806985731550830475727995

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity,a(n)/a(n-1)) = (2+sqrt(5))^8 = 51841+23184*sqrt(5). - Ant King, Dec 24 2011

Links

• Colin Barker, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Heptagonal Hexagonal Number
• Index entries for linear recurrences with constant coefficients, signature (103683,-103683,1).

Crossrefs

Cf. A048901, A048902.

Programs

Mathematica

LinearRecurrence[{103683, -103683, 1}, {1, 121771, 12625478965}, 8]; (* Ant King, Dec 24 2011 *)

PARI

Vec(-x*(55*x^2+18088*x+1)/((x-1)*(x^2-103682*x+1)) + O(x^20)) \\ Colin Barker, Jun 23 2015

Formula

From Ant King, Dec 24 2011: (Start)

G.f.: x*(1+18088*x+55*x^2)/((1-x)*(1-103682*x+x^2)).

a(n) = 103683*a(n-1)-103683*a(n-2)+a(n-3).

a(n) = 103682*a(n-1)-a(n-2)+18144.

a(n) = 1/80*((sqrt(5)-1)*(2+sqrt(5))^(8n-5)- (sqrt(5)+1)*(2-sqrt(5))^(8n-5)-14).

a(n) = floor(1/80*(sqrt(5)-1)*(2+sqrt(5))^(8n-5)).

(End)