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A006060 Triangular star numbers.
(Formerly M5425)
2

%I M5425

%S 1,253,49141,9533161,1849384153,358770992581,69599723176621,

%T 13501987525271953,2619315980179582321,508133798167313698381,

%U 98575337528478677903653,19123107346726696199610361

%N Triangular star numbers.

%D M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H B. Berselli, <a href="/A006060/b006060.txt">Table of n, a(n) for n = 1..400</a>. [From _Bruno Berselli_, Jul 07 2010]

%H Simon Plouffe, <a href="https://arxiv.org/ftp/arxiv/papers/0911/0911.4975.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarNumber.html">Star Number</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195, -195, 1).

%F G.f.: [1+58x+x^2]/[(x-1)(1-194x+x^2)]. - _Ralf Stephan_, Apr 23 2004

%F From _Bruno Berselli_, Jul 07 2010: (Start)

%F a(n) = 194*a(n-1)-a(n-2)+60 (n>2).

%F a(n) = (3*((7+4*3^(1/2))^(2*n-1)+(7-4*3^(1/2))^(2*n-1))-10)/32 (n>0).

%F (End)

%p A006060:=-(1+58*z+z**2)/(z-1)/(z**2-194*z+1); [Conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation.]

%p 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 = {}; Do[

%t If[Length[a006060] < 2, AppendTo[a006060, 1],

%t AppendTo[a006060, 194*a006060[[-1]] + 60 - a006060[[-2]]]], {n,

%t 20}]; TableForm[Transpose[List[Range[Length[a006060]], a006060]]] (* _Michael De Vlieger_ *)

%t LinearRecurrence[{195,-195,1},{1,253,49141},20] (* _Harvey P. Dale_, Jan 12 2017 *)

%Y Cf. A000567, A003154, A006061, A051673.

%K nonn,changed

%O 1,2

%A _N. J. A. Sloane_.

%E Extended by _Eric W. Weisstein_, Mar 01 2002

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Last modified March 3 09:46 EST 2021. Contains 341760 sequences. (Running on oeis4.)