OFFSET
1,2
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 60.
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
Bruno Berselli, Table of n, a(n) for n = 1..400.
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)/((1 - x)*(1 - 194x + x^2)). - Ralf Stephan, Apr 23 2004 [corrected by Stefano Spezia, Oct 18 2025]
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)
E.g.f.: exp(x)*(3*exp(96*x)*(7*cosh(56*sqrt(3)*x) + 4*sqrt(3)*sinh(56*sqrt(3)*x)) - 5)/16. - Stefano Spezia, Oct 18 2025
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Extended by Eric W. Weisstein, Mar 01 2002
STATUS
approved