A039835 Indices of triangular numbers which are also heptagonal.

1, 10, 493, 3382, 158905, 1089154, 51167077, 350704366, 16475640049, 112925716858, 5305104928861, 36361730124070, 1708227311453353, 11708364174233842, 550043889183050965, 3770056902373173214, 177112424089630957537, 1213946614199987541226

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

From Ant King, Oct 19 2011: (Start)

lim(n->Infinity,a(2n+1)/a(2n))=1/2(47+21*sqrt(5)).

lim(n->Infinity,a(2n)/a(2n-1))=1/2(7+3*sqrt(5)).

(End)

Links

• Colin Barker, Table of n, a(n) for n = 1..798
• Eric Weisstein's World of Mathematics, Heptagonal Triangular Number
• Index entries for linear recurrences with constant coefficients, signature (1,322,-322,-1,1).

Crossrefs

Cf. A046193, A046194.

Programs

Mathematica

LinearRecurrence[{1,322,-322,-1,1},{1,10,493,3382,158905},16] (* Ant King, Oct 19 2011 *)

PARI

Vec((-2*x^4-9*x^3+161*x^2+9*x+1)/((1-x)*(1-18*x+x^2)*(1+18*x+x^2))+O(x^99))

Formula

G.f.: (-2x^4-9x^3+161x^2+9x+1)/[(1-x)(1-18x+x^2)(1+18x+x^2)].

a(n+2) = 322*a(n+1)-a(n)+160 a(n+1) = 161*a(n)+80+36*(20*a(n)^2+20*a(n)+9)^0.5 - Richard Choulet, Sep 29 2007

From Ant King, Oct 19 2011: (Start)

a(n) = a(n-1)+322a(n-2)-322a(n-3)-a(n-4)+a(n-5).

a(n) = 1/20*sqrt(5)*(( sqrt(5)-(-1)^n)*(2+ sqrt(5))^(2n-1)+( sqrt(5)+(-1)^n)*(2- sqrt(5))^(2n-1)-2* sqrt(5)).

a(n) = floor(1/20* sqrt(5)*(sqrt(5)-(-1)^n)*(2+ sqrt(5))^(2n-1))(End)