A015520 a(n) = 2*a(n-1) + 11*a(n-2), a(0) = 0, a(1) = 1.

0, 1, 2, 15, 52, 269, 1110, 5179, 22568, 102105, 452458, 2028071, 9033180, 40375141, 180115262, 804357075, 3589982032, 16027891889, 71545586130, 319397983039, 1425797413508, 6364972640445, 28413716829478

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A090042.

Programs

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

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 *)

PARI

x='x+O('x^30); concat([0], Vec(-x/(-1+2*x+11*x^2))) \\ G. C. Greubel, Jan 01 2018

Sage

[lucas_number1(n,2,-11) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009

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