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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved