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A046174
Indices of pentagonal numbers which are also triangular.
10
0, 1, 12, 165, 2296, 31977, 445380, 6203341, 86401392, 1203416145, 16761424636, 233456528757, 3251629977960, 45289363162681, 630799454299572, 8785902997031325, 122371842504138976, 1704419892060914337
OFFSET
0,3
LINKS
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.
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
MATHEMATICA
LinearRecurrence[{15, -15, 1}, {0, 1, 12}, 20] (* Harvey P. Dale, Aug 22 2011 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
STATUS
approved