A046180 Hexagonal pentagonal numbers.

1, 40755, 1533776805, 57722156241751, 2172315626468283465, 81752926228785223683195, 3076689623521787481625080301, 115788137209866023854693048367775, 4357570752679408318225730700647767185, 163992817590548715438241125333485021875651

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(3))^8 = 18817 + 10864*sqrt(3). - Ant King, Dec 13 2011

Dickson calls the terms "triangular, pentagonal and hexagonal" (all hexagonal numbers are also triangular). - Jonathan Sondow, May 06 2014

Links

• Colin Barker, Table of n, a(n) for n = 1..219
• L. E. Dickson, History of the Theory of Numbers, vol. II, pp. 19-20.
• Eric Weisstein's World of Mathematics, Hexagonal Pentagonal Number.
• Index entries for linear recurrences with constant coefficients, signature (37635,-37635,1).

Crossrefs

Cf. A046178, A046179.

Programs

Mathematica

LinearRecurrence[{37635, -37635, 1}, {1, 40755, 1533776805}, 8] (* Ant King, Dec 13 2011 *)

PARI

Vec(x*(1+3120*x+15*x^2)/((1-x)*(1-37634*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 21 2015

Formula

a(n) = 37634*a(n-1) - a(n-2) + 3136; g.f.: x*(1+3120*x+15*x^2)/((1-x)*(1-37634*x+x^2)). - Warut Roonguthai Jan 08 2001

a(n+1) = 18817*a(n)+1568+1358*(192*a(n)^2+32*a(n)+1)^0.5 - Richard Choulet, Sep 19 2007

From Ant King, Dec 13 2011: (Start)

a(n) = 37635*a(n-1) - 37635*a(n-2) + a(n-3).

a(n) = (1/48)*((2+sqrt(3))^(8n-5)+(2-sqrt(3))^(8n-5)-4).

a(n) = floor((1/48)*(2+sqrt(3))^(8n-5)).

a(n) = (1/48)*((tan(5*Pi/12))^(8n-5)+(tan(Pi/12))^(8n-5)-4).

a(n) = floor((1/48)*(tan(5*pi/12))^(8n-5)).

(End)