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

1,2

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).

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)

Mathematica

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

Prog

(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))

Crossrefs

Cf. A046193, A046194.

Sequence in context: A200458 A223122 A071096 * A293090 A127947 A095232

Adjacent sequences: A039832 A039833 A039834 * A039836 A039837 A039838

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved