A006060 Triangular star numbers.

1, 253, 49141, 9533161, 1849384153, 358770992581, 69599723176621, 13501987525271953, 2619315980179582321, 508133798167313698381, 98575337528478677903653, 19123107346726696199610361

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• B. Berselli, Table of n, a(n) for n = 1..400. [From _Bruno Berselli_, Jul 07 2010]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Eric Weisstein's World of Mathematics, Star Number
• Index entries for linear recurrences with constant coefficients, signature (195, -195, 1).

Crossrefs

Cf. A000567, A003154, A006061, A051673.

Programs

Maple

A006060:=-(1+58*z+z**2)/(z-1)/(z**2-194*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
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

Mathematica

a006060 = {}; Do[
If[Length[a006060] < 2, AppendTo[a006060, 1],
  AppendTo[a006060, 194*a006060[[-1]] + 60 - a006060[[-2]]]],  {n,
  20}]; TableForm[Transpose[List[Range[Length[a006060]], a006060]]] (* Michael De Vlieger *)
LinearRecurrence[{195,-195,1},{1,253,49141},20] (* Harvey P. Dale, Jan 12 2017 *)

Formula

G.f.: (1 + 58x + x^2)/((x-1)(1 - 194x + x^2)). - Ralf Stephan, Apr 23 2004

From Bruno Berselli, Jul 07 2010: (Start)

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

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

(End)

Extensions

Extended by Eric W. Weisstein, Mar 01 2002