A006060 Triangular star numbers. (Formerly M5425)

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

Offset

1,2

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).

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)

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 *)

Crossrefs

Cf. A000567, A003154, A006061, A051673.

Sequence in context: A271985 A255182 A176087 * A077695 A253880 A145628

Adjacent sequences: A006057 A006058 A006059 * A006061 A006062 A006063

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended by Eric W. Weisstein, Mar 01 2002

Status

approved