A046174 Indices of pentagonal numbers which are also triangular.
0, 1, 12, 165, 2296, 31977, 445380, 6203341, 86401392, 1203416145, 16761424636, 233456528757, 3251629977960, 45289363162681, 630799454299572, 8785902997031325, 122371842504138976, 1704419892060914337
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Roy. Sci. Liège 33 (1964) 513-517.
- W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Roy. Sci. Liège 33 (1964) 513-517.
- Eric Weisstein's World of Mathematics, Pentagonal Triangular Number.
- Index entries for linear recurrences with constant coefficients, signature (15,-15,1)
Programs
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Magma
[ n eq 1 select 0 else n eq 2 select 1 else 14*Self(n-1)-Self(n-2)-2: n in [1..20] ]; // Vincenzo Librandi, Aug 23 2011
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Mathematica
LinearRecurrence[{15,-15,1},{0,1,12},20] (* Harvey P. Dale, Aug 22 2011 *)
Formula
From Warut Roonguthai, Jan 05 2001: (Start)
a(n) = 14*a(n-1) - a(n-2) - 2.
G.f.: x*(1-3*x)/((1-x)*(1-14*x+x^2)). (End)
a(n+1) = 7*a(n) - 1 + 2*sqrt(12*a(n)^2 - 4*a(n) + 1). - Richard Choulet, Sep 19 2007
a(n+1) = 15*a(n) - 15*a(n-1) + a(n-2), a(1)=1, a(2)=12, a(3)=165. - Sture Sjöstedt, May 29 2009
a(n) = (1/12)*(2 - (7 - 4*sqrt(3))^n*(1 + sqrt(3)) + (-1 + sqrt(3))*(7 + 4*sqrt(3))^n). - Alan Michael Gómez Calderón, Jun 30 2024