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A006603 Generalized Fibonacci numbers.

%I A006603 M1771 #34 Oct 28 2024 04:52:45
%S A006603 1,2,7,26,107,468,2141,10124,49101,242934,1221427,6222838,32056215,
%T A006603 166690696,873798681,4612654808,24499322137,130830894666,702037771647,
%U A006603 3783431872018,20469182526595,111133368084892,605312629105205,3306633429423460,18111655081108453
%N A006603 Generalized Fibonacci numbers.
%C A006603 The Kn21 sums, see A180662, of the Schroeder triangle A033877 equal A006603(n) while the Kn3 sums equal A006603(2*n). The Kn22 sums, see A227504, and the Kn23 sums, see A227505, are also related to the sequence given above. - _Johannes W. Meijer_, Jul 15 2013
%C A006603 Typo on the right-hand side of Rogers's equation (1-x+x^2+x^3)*R^*(x) = R(x) + x: the sign in front of the x should be switched. - _R. J. Mathar_, Nov 23 2018
%D A006603 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006603 G. C. Greubel, <a href="/A006603/b006603.txt">Table of n, a(n) for n = 0..1000</a>
%H A006603 D. G. Rogers, <a href="https://dx.doi.org/10.1007/BFb0069192">A Schroeder triangle: three combinatorial problems</a>, in "Combinatorial Mathematics V (Melbourne 1976)", Lect. Notes Math. 622 (1976), pp. 175-196.
%F A006603 a(n) = abs(A080244(n-1)).
%F A006603 G.f.: (1 - x - 2*x^2 - sqrt(1 - 6*x + x^2))/(2*x*(1 - x + x^2 + x^3)).
%F A006603 G.f.: (A006318(x) - x)/(1 - x + x^2 + x^3).
%F A006603 a(n) = Sum_{k=1..floor(n/2)+1} k*(1/(n-k+2))*Sum_{i=0..n-2*k+2} C(n-k+2,i)*C(2*n-3*k-i+3,n-k+1). - _Vladimir Kruchinin_, Oct 23 2011
%F A006603 (n+1)*a(n) -(7*n-2)*a(n-1) +4*(2*n-1)*a(n-2) -6*(n-1)*a(n-3) -(5*n-1)*a(n-4) +(n-2)*a(n-5) = 0. - _R. J. Mathar_, Nov 23 2018
%p A006603 A006603 := n-> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): seq(A006603(n), n=0..24); # _Johannes W. Meijer_, Jul 15 2013
%t A006603 CoefficientList[Series[(1-x-2x^2-Sqrt[1-6x+x^2])/(2x(1-x+x^2+x^3)),{x,0,30}],x] (* _Harvey P. Dale_, Jun 12 2016 *)
%o A006603 (Maxima) a(n):=sum((k*sum(binomial(n-k+2,i)*binomial(2*n-3*k-i+3,n-k+1),i,0,n-2*k+2))/(n-k+2),k,1,n/2+1); /* _Vladimir Kruchinin_, Oct 23 2011 */
%o A006603 (Magma)
%o A006603 R<x>:=PowerSeriesRing(Rationals(), 50);
%o A006603 Coefficients(R!( (1-x-2*x^2 -Sqrt(1-6*x+x^2))/(2*x*(1-x+x^2+x^3)) )); // _G. C. Greubel_, Oct 27 2024
%o A006603 (SageMath)
%o A006603 def A006603_list(prec):
%o A006603     P.<x> = PowerSeriesRing(ZZ, prec)
%o A006603     return P( (1-x-2*x^2 -sqrt(1-6*x+x^2))/(2*x*(1-x+x^2+x^3)) ).list()
%o A006603 A006603_list(50) # _G. C. Greubel_, Oct 27 2024
%Y A006603 Cf. A006318, A006603, A033877, A080244, A180662, A227504, A227505.
%K A006603 nonn,easy
%O A006603 0,2
%A A006603 _N. J. A. Sloane_, _Simon Plouffe_
%E A006603 More terms from _Emeric Deutsch_, Feb 28 2004