A046175 Indices of triangular numbers which are also pentagonal.
0, 1, 20, 285, 3976, 55385, 771420, 10744501, 149651600, 2084377905, 29031639076, 404358569165, 5631988329240, 78443478040201, 1092576704233580, 15217630381229925, 211954248632985376, 2952141850480565345, 41118031658094929460, 572700301362848447101
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..874
- W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Roy. Sci. Liege 33 (1964) 513-517
- Eric Weisstein's World of Mathematics, Pentagonal Triangular Number, MathWorld
- Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
Programs
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Mathematica
LinearRecurrence[{15,-15,1},{0,1,20},20] (* Harvey P. Dale, Sep 10 2021 *) -
PARI
concat(0, Vec(-x*(5*x+1)/((x-1)*(x^2-14*x+1)) + O(x^50))) \\ Colin Barker, Jun 23 2015
Formula
From Warut Roonguthai, Jan 05 2001: (Start)
a(n) = 14*a(n-1) - a(n-2) + 6.
G.f.: x*(1+5*x)/((1-x)*(1-14*x+x^2)). (End)
a(n+1) = 7*a(n) + 3 + 2*sqrt(12*a(n)^2 + 12*a(n) + 1). - Richard Choulet, Sep 19 2007
a(n+1) = 15*a(n)-15*a(n-1)+ a(n-2) with a(1)=1, a(2)=20, a(3)=285. - Sture Sjöstedt, May 29 2009
a(n) = (1/12)*(-6 + (7 - 4*sqrt(3))^n*(3 + sqrt(3)) - (-3 + sqrt(3))*(7 + 4*sqrt(3))^n). - Alan Michael Gómez Calderón, Jun 30 2024