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A006604 Generalized Fibonacci numbers.
(Formerly M3469)
1

%I M3469

%S 1,1,4,13,53,228,1037,4885,23640,116793,586633,2986616,15377097,

%T 79927913,418852716,2210503285,11738292397,62673984492,336260313765,

%U 1811960161517,9802082905840,53213718977777,289817858570513,1583076422786096,8670574105626961

%N Generalized Fibonacci numbers.

%D D. G. Rogers, A Schroeder triangle: three combinatorial problems, in "Combinatorial Mathematics V (Melbourne 1976)", Lect. Notes Math. 622 (1976), pp. 175-196.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Plouffe, Simon, <a href="http://arxiv.org/abs/0911.4975">Approximations of generating functions and a few conjectures</a>, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.

%F G.f.: (1+x-2*x^2-sqrt(1-6*x+x^2))/(2*(2*x-x^2-x^3+x^4)).

%F n*a(n) = (-1/2*n+3/2)*a(n-5)+(7/2*n-6)*a(n-4) +(13/2*n-9)*a(n-1) +(-7/2*n+15/2) *a(n-2) +(-3*n+3)*a(n-3). - _Simon Plouffe_, Master's Thesis, UQAM, 1992

%F a(n) = sum(k=1..n/2+1, (k*sum(j=0..n-2*k+2, (-1)^j*2^(n-2*k-j+2)*C(n-k+2,j) * C(2*n-3*k-j+3,n-k+1)))/((n-k+2))). - _Vladimir Kruchinin_, Oct 22 2011

%t CoefficientList[Series[(1+x-2 x^2-Sqrt[1-6 x+x^2])/(2 (2 x-x^2-x^3+x^4)),{x,0,60}], x] (* _Harvey P. Dale_, Mar 23 2011 *)

%o (Maxima) a(n):=sum((k*sum((-1)^j*2^(n-2*k-j+2)*binomial(n-k+2,j)*binomial(2*n-3*k-j+3,n-k+1),j,0,n-2*k+2))/((n-k+2)),k,1,n/2+1); // _Vladimir Kruchinin_, Oct 22 2011

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Harvey P. Dale_, Mar 23 2011

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Last modified July 29 01:22 EDT 2021. Contains 346340 sequences. (Running on oeis4.)