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%I #54 Mar 18 2023 05:35:19
%S 1,1,12,23,155,408,2113,6601,29844,102455,430739,1557744,6295873,
%T 23431057,92685660,350427287,1369969547,5224669704,20294334721,
%U 77765701465,301003383396,1156426099511,4467463316867,17188150411488
%N Generalized Fibonacci numbers: a(n) = a(n-1) + 11*a(n-2).
%C 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
%H Vincenzo Librandi, <a href="/A015447/b015447.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,11).
%F a(n) = ( ( (1+3*sqrt(5))/2 )^(n+1) - ( (1-3*sqrt(5))/2 )^(n+1) )/(3*sqrt(5)).
%F a(n-1) = (1/3)*(-1)^n*Sum_{i=0..n} (-3)^i*Fibonacci(i)*C(n, i). - _Benoit Cloitre_, Mar 06 2004
%F a(n) = Sum_{k=0..n} A109466(n,k)*(-11)^(n-k). - _Philippe Deléham_, Oct 26 2008
%F G.f.: 1/(1 - x - 11*x^2). - _Harvey P. Dale_, May 08 2011
%F a(n) = ( Sum_{1<=k<=n+1, k odd} C(n+1,k)*45^((k-1)/2) )/2^n. - _Vladimir Shevelev_, Feb 05 2014
%t Join[{a=1,b=1},Table[c=b+11*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)
%t 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 *)
%o (Sage) [lucas_number1(n,1,-11) for n in range(0, 27)] # _Zerinvary Lajos_, Apr 22 2009
%o (Magma) [n le 2 select 1 else Self(n-1) + 11*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 06 2012
%o (PARI) Vec(1/(1-x-11*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Feb 03 2014
%Y Cf. A015446, A015443.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
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