A048901 Indices of hexagonal numbers which are also heptagonal.

1, 247, 79453, 25583539, 8237820025, 2652552464431, 854113655726677, 275021944591525483, 88556212044815478769, 28514825256485992638055, 9181685176376444813974861, 2956474111967958744107267107

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))^4 = 161 + 72*sqrt(5). - Ant King, Dec 24 2011

Links

• Vincenzo Librandi, 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 (323,-323,1).

Crossrefs

Cf. A048902, A048903.

Programs

Magma

I:=[1, 247, 79453]; [n le 3 select I[n] else 323*Self(n-1)-323*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Dec 28 2011

Mathematica

LinearRecurrence[{323, -323, 1}, {1, 247, 79453}, 12]; (* Ant King, Dec 24 2011 *)

Formula

G.f.: x*(-1 + 76*x + 5*x^2) / ( (x-1)*(x^2 - 322*x + 1) ). - R. J. Mathar, Dec 21 2011

From Ant King, Dec 24 2011: (Start)

a(n) = 322*a(n-1) - a(n-2) - 80.

a(n) = (1/40)*sqrt(5)*((1+sqrt(5))*(sqrt(5)+2)^(4*n-3) + (1-sqrt(5))*(sqrt(5)-2)^(4*n-3) + 2*sqrt(5)).

a(n) = ceiling((1/40)*sqrt(5)*(1+sqrt(5))*(sqrt(5)+2)^(4*n-3)).

(End)