OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (2, 11).
FORMULA
a(n) = 2*a(n-1) + 11*a(n-2).
E.g.f.: exp(x)*sinh(2*sqrt(3)*x)/(2*sqrt(3)). - Paul Barry, Nov 20 2003
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*12^k. - Paul Barry, Sep 29 2004
From R. J. Mathar, Apr 29 2008: (Start)
O.g.f.: x/(1 - 2*x - 11*x^2).
a(n) = 11^n*(A^n - B^n)/(4*sqrt(3)) where A = 1/(2*sqrt(3)-1) and B = -1/(2*sqrt(3)+1). (End)
a(n) = (Ap^n - Am^n)/(Ap - Am), where Ap = 1 + 2*sqrt(3) and Am = 1 - 2*sqrt(3). (Binet - de Moivre type formula.) This coincides with the preceding formula. - Wolfdieter Lang, Feb 17 2018
MATHEMATICA
Join[{a=0, b=1}, Table[c=2*b+11*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{2, 11}, {0, 1}, 30] (* Harvey P. Dale, Jul 13 2011 *)
PROG
(Sage) [lucas_number1(n, 2, -11) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009
(Magma) [ n eq 1 select 0 else n eq 2 select 1 else 2*Self(n-1)+11*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011
(PARI) x='x+O('x^30); concat([0], Vec(-x/(-1+2*x+11*x^2))) \\ G. C. Greubel, Jan 01 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved