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A046174 Indices of pentagonal numbers which are also triangular.

%I A046174 #45 Feb 16 2025 08:32:39
%S A046174 0,1,12,165,2296,31977,445380,6203341,86401392,1203416145,16761424636,
%T A046174 233456528757,3251629977960,45289363162681,630799454299572,
%U A046174 8785902997031325,122371842504138976,1704419892060914337
%N A046174 Indices of pentagonal numbers which are also triangular.
%H A046174 Vincenzo Librandi, <a href="/A046174/b046174.txt">Table of n, a(n) for n = 0..500</a>
%H A046174 W. Sierpiński, <a href="https://popups.uliege.be/0037-9565/index.php?id=3612">Sur les nombres pentagonaux</a>, Bull. Soc. Roy. Sci. Liège 33 (1964) 513-517.
%H A046174 W. Sierpiński, <a href="http://www.ams.org/mathscinet-getitem?mr=171743">Sur les nombres pentagonaux</a>, Bull. Soc. Roy. Sci. Liège 33 (1964) 513-517.
%H A046174 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalTriangularNumber.html">Pentagonal Triangular Number.</a>
%H A046174 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1)
%F A046174 From _Warut Roonguthai_, Jan 05 2001: (Start)
%F A046174 a(n) = 14*a(n-1) - a(n-2) - 2.
%F A046174 G.f.: x*(1-3*x)/((1-x)*(1-14*x+x^2)). (End)
%F A046174 a(n+1) = 7*a(n) - 1 + 2*sqrt(12*a(n)^2 - 4*a(n) + 1). - _Richard Choulet_, Sep 19 2007
%F A046174 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
%F A046174 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
%t A046174 LinearRecurrence[{15,-15,1},{0,1,12},20] (* _Harvey P. Dale_, Aug 22 2011 *)
%o A046174 (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
%Y A046174 Cf. A014979, A046175, A001834.
%K A046174 nonn,easy
%O A046174 0,3
%A A046174 _Eric W. Weisstein_