OFFSET
1,2
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 121, p. 42, Ellipses, Paris 2008.
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. 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..500
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 (99,-99,1).
FORMULA
a(n) = denominator of kappa(sqrt(6)/A054320(n)) where kappa(x) is the sum of successive remainders by computing the Euclidean algorithm for (1, x). - Thomas Baruchel, Nov 29 2003
From Ignacio Larrosa Cañestro, Feb 27 2000: (Start)
a(n) = 99*(a(n-1) - a(n-2)) + a(n-3).
a(n) = (5 - 2*sqrt(6))/8*(sqrt(3) + sqrt(2))^(4*n) + (5 + 2*sqrt(6))/8*(sqrt(3) - sqrt(2))^(4*n) - 1/4. (End)
a(n) = 98*a(n-1) - a(n-2) + 24. - Lekraj Beedassy, Jul 14 2008
E.g.f.: exp(x)*(exp(48*x)*(5*cosh(20*sqrt(6)*x) - 2*sqrt(6)*sinh(20*sqrt(6)*x)) - 1)/4 - 1. - Stefano Spezia, Oct 18 2025
EXAMPLE
a(2)=121 because this is the 2nd star number (A003154) that is a square.
MAPLE
Digits := 1000:q := seq(floor(evalf(( (5+2*sqrt(6))^n*(sqrt(6)-2)-(5-2*sqrt(6))^n*(sqrt(6)+2))^2/16)), n=1..100);
A006061:=-(1+22*z+z**2)/(z-1)/(z**2-98*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
MATHEMATICA
CoefficientList[Series[(1+22*x+x^2)/((1-x)*(1-98*x+x^2)), {x, 0, 20}], x] (* G. C. Greubel, Jul 23 2019 *)
(* Alternative: *)
LinearRecurrence[{99, -99, 1}, {1, 121, 11881}, 20] (* G. C. Greubel, Jul 23 2019 *)
PROG
(PARI) my(x='x+O('x^20)); Vec((1+22*x+x^2)/((1-x)*(1-98*x+x^2))) \\ G. C. Greubel, Jul 23 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+22*x+x^2)/((1-x)*(1-98*x+x^2)) )); // G. C. Greubel, Jul 23 2019
(SageMath) ((1+22*x+x^2)/((1-x)*(1-98*x+x^2))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019
(GAP) a:=[1, 121, 11881];; for n in [4..20] do a[n]:=99*a[n-1]-99*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jul 23 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Eric W. Weisstein and Sascha Kurz, Mar 24 2002
STATUS
approved