

A015447


Generalized Fibonacci numbers: a(n) = a(n1) + 11*a(n2).


21



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

0,3


COMMENTS

The compositions of n in which each positive integer is colored by one of p different colors are called pcolored compositions of n. For n>=2, 12*a(n2) equals the number of 12colored 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)  ( (13*sqrt(5))/2 )^(n+1) )/3*sqrt(5).
a(n1) = (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)^(nk).  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^((k1)/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/(1x11 x^2), {x, 0, 50}], x] (* Harvey P. Dale, May 08 2011 *)


PROG

(Sage) [lucas_number1(n, 1, 11) for n in xrange(0, 27)] # Zerinvary Lajos, Apr 22 2009
(MAGMA) [n le 2 select 1 else Self(n1) + 11*Self(n2): n in [1..30]]; // Vincenzo Librandi, Nov 06 2012
(PARI) Vec(1/(1x11*x^2)+O(x^99)) \\ Charles R Greathouse IV, Feb 03 2014


CROSSREFS

Cf. A015446, A015443.
Sequence in context: A083683 A294139 A255766 * A072822 A239656 A059161
Adjacent sequences: A015444 A015445 A015446 * A015448 A015449 A015450


KEYWORD

nonn,easy


AUTHOR

Olivier Gérard


STATUS

approved



