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A006604 Generalized Fibonacci numbers.

%I A006604 M3469 #40 Aug 30 2025 05:08:09
%S A006604 1,1,4,13,53,228,1037,4885,23640,116793,586633,2986616,15377097,
%T A006604 79927913,418852716,2210503285,11738292397,62673984492,336260313765,
%U A006604 1811960161517,9802082905840,53213718977777,289817858570513,1583076422786096,8670574105626961
%N A006604 Generalized Fibonacci numbers.
%D A006604 D. G. Rogers, A Schroeder triangle: three combinatorial problems, in "Combinatorial Mathematics V (Melbourne 1976)", Lect. Notes Math. 622 (1976), pp. 175-196.
%D A006604 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006604 G. C. Greubel, <a href="/A006604/b006604.txt">Table of n, a(n) for n = 0..1000</a>
%H A006604 Simon Plouffe, <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 A006604 G.f.: (1+x-2*x^2-sqrt(1-6*x+x^2))/(2*(2*x-x^2-x^3+x^4)).
%F A006604 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 A006604 a(n) = Sum_{k=1..n/2+1} (k/(n-k+2))*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). - _Vladimir Kruchinin_, Oct 22 2011
%F A006604 G.f.: (1 - x + f(x))/(2 - x - x^2 + x^3), where f(x) is the gf of A006318. - _G. C. Greubel_, Aug 29 2025
%t A006604 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 A006604 (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 */
%o A006604 (Magma)
%o A006604 R<x>:=PowerSeriesRing(Rationals(), 40);
%o A006604 Coefficients(R!( (1+x-2*x^2-Sqrt(1-6*x+x^2))/(2*(2*x-x^2-x^3+x^4)) )); // _G. C. Greubel_, Aug 29 2025
%o A006604 (SageMath)
%o A006604 def A006604_list(prec):
%o A006604     P.<x>= PowerSeriesRing(QQ, prec)
%o A006604     return P( (1+x-2*x^2-sqrt(1-6*x+x^2))/(2*(2*x-x^2-x^3+x^4)) ).list()
%o A006604 print(A006604_list(41)) # _G. C. Greubel_, Aug 29 2025
%Y A006604 Cf. A006318, A006603.
%K A006604 nonn,changed
%O A006604 0,3
%A A006604 _N. J. A. Sloane_
%E A006604 More terms from _Harvey P. Dale_, Mar 23 2011