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A006060 Triangular star numbers.

%I A006060 M5425 #45 Feb 16 2025 08:32:29
%S A006060 1,253,49141,9533161,1849384153,358770992581,69599723176621,
%T A006060 13501987525271953,2619315980179582321,508133798167313698381,
%U A006060 98575337528478677903653,19123107346726696199610361
%N A006060 Triangular star numbers.
%D A006060 M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
%D A006060 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006060 B. Berselli, <a href="/A006060/b006060.txt">Table of n, a(n) for n = 1..400</a>. [From _Bruno Berselli_, Jul 07 2010]
%H A006060 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A006060 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006060 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StarNumber.html">Star Number</a>
%H A006060 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195, -195, 1).
%F A006060 G.f.: (1 + 58x + x^2)/((x-1)(1 - 194x + x^2)). - _Ralf Stephan_, Apr 23 2004
%F A006060 From _Bruno Berselli_, Jul 07 2010: (Start)
%F A006060 a(n) = 194*a(n-1) - a(n-2) + 60 (n>2).
%F A006060 a(n) = (3*((7 + 4*sqrt(3))^(2*n-1) + (7 - 4*sqrt(3))^(2*n-1)) - 10)/32 (n>0).
%F A006060 (End)
%p A006060 A006060:=-(1+58*z+z**2)/(z-1)/(z**2-194*z+1); # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation
%p A006060 a:= n-> (Matrix([[253,1,1]]). Matrix([[195,1,0], [ -195,0,1], [1,0,0]])^n)[1,3]: seq(a(n), n=1..20); # _Alois P. Heinz_, Aug 14 2008
%t A006060 a006060 = {}; Do[
%t A006060 If[Length[a006060] < 2, AppendTo[a006060, 1],
%t A006060   AppendTo[a006060, 194*a006060[[-1]] + 60 - a006060[[-2]]]],  {n,
%t A006060   20}]; TableForm[Transpose[List[Range[Length[a006060]], a006060]]] (* _Michael De Vlieger_ *)
%t A006060 LinearRecurrence[{195,-195,1},{1,253,49141},20] (* _Harvey P. Dale_, Jan 12 2017 *)
%Y A006060 Cf. A000567, A003154, A006061, A051673.
%K A006060 nonn
%O A006060 1,2
%A A006060 _N. J. A. Sloane_
%E A006060 Extended by _Eric W. Weisstein_, Mar 01 2002