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URL: https://oeis.org/A046180

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A046180
Hexagonal pentagonal numbers.
5
1, 40755, 1533776805, 57722156241751, 2172315626468283465, 81752926228785223683195, 3076689623521787481625080301, 115788137209866023854693048367775, 4357570752679408318225730700647767185, 163992817590548715438241125333485021875651, 6171705692845139604123358192574644612620485685
OFFSET
1,2
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
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 37.
LINKS
L. E. Dickson, History of the Theory of Numbers, vol. II, pp. 19-20.
Eric Weisstein's World of Mathematics, Hexagonal Pentagonal Number.
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)
MATHEMATICA
LinearRecurrence[{37635, -37635, 1}, {1, 40755, 1533776805}, 8] (* Ant King, Dec 13 2011 *)
PROG
(PARI) Vec(x*(1+3120*x+15*x^2)/((1-x)*(1-37634*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 21 2015
CROSSREFS
Sequence in context: A097238 A384400 A249879 * A031668 A164648 A232301
KEYWORD
nonn,easy
STATUS
approved