A048904 Indices of heptagonal numbers (A000566) which are also octagonal.

1, 345, 166145, 80081401, 38599068993, 18604671173081, 8967412906355905, 4322274416192372985, 2083327301191817422721, 1004159436900039805378393, 484002765258517994374962561, 233288328695168773248926575865

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) = (sqrt(5) + sqrt(6))^4 = 241 + 44*sqrt(30). - Ant King, Dec 30 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Octagonal Heptagonal Number.
• Index entries for linear recurrences with constant coefficients, signature (483,-483,1).

Crossrefs

Cf. A048905, A048906.

Programs

Magma

I:=[1, 345, 166145]; [n le 3 select I[n] else 483*Self(n-1)-483*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Dec 28 2011

Mathematica

LinearRecurrence[{483,-483,1},{1,345,166145},30]

Formula

G.f.: x*(-1 + 138*x + 7*x^2) / ( (x-1)*(x^2 - 482*x + 1) ). - R. J. Mathar, Dec 21 2011

From Ant King, Dec 30 2011: (Start)

a(n) = 482*a(n-1) - a(n-2) - 144.

a(n) = (1/60)*((3*sqrt(5) + sqrt(6))*(sqrt(5) + sqrt(6))^(4*n-3) + (3*sqrt(5) - sqrt(6))*(sqrt(5) - sqrt(6))^(4*n-3) + 18).

a(n) = ceiling((1/60)*(3*sqrt(5) + sqrt(6))*(sqrt(5) + sqrt(6))^(4*n-3)). (End)