A015447 Generalized Fibonacci numbers: a(n) = a(n-1) + 11*a(n-2).

1, 1, 12, 23, 155, 408, 2113, 6601, 29844, 102455, 430739, 1557744, 6295873, 23431057, 92685660, 350427287, 1369969547, 5224669704, 20294334721, 77765701465, 301003383396, 1156426099511, 4467463316867, 17188150411488

Offset

0,3

Comments

The compositions of n in which each positive integer is colored by one of p different colors are called p-colored compositions of n. For n>=2, 12*a(n-2) equals the number of 12-colored compositions of n, with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,11).

Formula

a(n) = ( ( (1+3*sqrt(5))/2 )^(n+1) - ( (1-3*sqrt(5))/2 )^(n+1) )/(3*sqrt(5)).

a(n-1) = (1/3)*(-1)^n*Sum_{i=0..n} (-3)^i*Fibonacci(i)*C(n, i). - Benoit Cloitre, Mar 06 2004

a(n) = Sum_{k=0..n} A109466(n,k)*(-11)^(n-k). - Philippe Deléham, Oct 26 2008

G.f.: 1/(1 - x - 11*x^2). - Harvey P. Dale, May 08 2011

a(n) = ( Sum_{1<=k<=n+1, k odd} C(n+1,k)*45^((k-1)/2) )/2^n. - Vladimir Shevelev, Feb 05 2014

Mathematica

Join[{a=1, b=1}, Table[c=b+11*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{1, 11}, {1, 1}, 30] (* or *) CoefficientList[Series[ 1/(1-x-11 x^2), {x, 0, 50}], x] (* Harvey P. Dale, May 08 2011 *)

Prog

(Sage) [lucas_number1(n, 1, -11) for n in range(0, 27)] # Zerinvary Lajos, Apr 22 2009
(Magma) [n le 2 select 1 else Self(n-1) + 11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 06 2012
(PARI) Vec(1/(1-x-11*x^2)+O(x^99)) \\ Charles R Greathouse IV, Feb 03 2014

Crossrefs

Cf. A015446, A015443.

Sequence in context: A294139 A255766 A333933 * A072822 A239656 A059161

Adjacent sequences: A015444 A015445 A015446 * A015448 A015449 A015450

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved