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A263878 a(n) = Sum_{k=0..n} (-1)^k*k*Fibonacci(k), where Fibonacci(k) = A000045(k).

%I A263878 #17 Feb 16 2025 08:33:27
%S A263878 0,-1,1,-5,7,-18,30,-61,107,-199,351,-628,1100,-1929,3349,-5801,9991,
%T A263878 -17158,29354,-50085,85215,-144651,244991,-414120,698712,-1176913,
%U A263878 1979305,-3323981,5574727,-9337914,15623286,-26111053,43594835,-72716239,121181919,-201779356
%N A263878 a(n) = Sum_{k=0..n} (-1)^k*k*Fibonacci(k), where Fibonacci(k) = A000045(k).
%H A263878 Colin Barker, <a href="/A263878/b263878.txt">Table of n, a(n) for n = 0..1000</a>
%H A263878 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%H A263878 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,1,-3,1).
%F A263878 a(n) = (-1)^n*(F(n-3) + n*F(n-1)) - 2, where F(n) = A000045(n).
%F A263878 G.f.: x*(x^2+1)/((x-1)*(x^2-x-1)^2).
%F A263878 E.g.f.: (exp(x/phi)*(phi^3+x)+exp(-phi*x)*(1/phi^3-x))/sqrt(5)-2*exp(x), where phi=(1+sqrt(5))/2.
%F A263878 Recurrences:
%F A263878 6-term, homogeneous, constant coefficients: a(0) = 0, a(1) = -1, a(2) = 1, a(3) = -5, a(4) = 7, a(n) = -a(n-1) + 3*a(n-2) + a(n-3) - 3*a(n-4) + a(n-5).
%F A263878 5-term, non-homogeneous, constant coefficients: a(0) = 0, a(1) = -1, a(2) = 1, a(3) = -5, a(n) = -2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4) - 2.
%F A263878 4-term, homogeneous: a(0) = 0, a(1) = -1, a(2) = 1, (n-1)*(n-2)*a(n) = (2-n)*a(n-1) + n*(2*n-3)*a(n-2) + n*(1-n)*a(n-3).
%F A263878 3-term, non-homogeneous: a(0) = 0, a(1) = -1, (n^2-1)*a(n) = -(n^2+n+1)*a(n-1) + n*(n+2)*a(n-2) - 2*n*(n-1).
%F A263878 0 = a(n)*(-2*a(n) + 15*a(n+1) - 9*a(n+2) + a(n+3) - 3*a(n+4)) + a(n+1)*(-25*a(n+1) + 15*a(n+2) + 15*a(n+3) + 5*a(n+4)) + a(n+2)*(18*a(n+2) - 29*a(n+3) - 13*a(n+4)) + a(n+3)*(+3*a(n+3) + 7*a(n+4)) + a(n+4)*(2*a(n+4)) for all n in Z. - _Michael Somos_, Nov 02 2015
%e A263878 G.f. = - x + x^2 - 5*x^3 + 7*x^4 - 18*x^5 + 30*x^6 - 61*x^7 + 107*x^8 - 199*x^9 + ...
%t A263878 Table[Sum[(-1)^k k Fibonacci[k], {k, 0, n}], {n, 0, 20}]
%t A263878 Table[(-1)^n (Fibonacci[n-3] + n Fibonacci[n-1]) - 2, {n, 0, 20}]
%o A263878 (PARI) concat(0, Vec(x*(x^2+1)/((x-1)*(x^2-x-1)^2) + O(x^40))) \\ _Colin Barker_, Oct 31 2015
%o A263878 (PARI) a(n) = (-1)^n*(fibonacci(n-3) + n*fibonacci(n-1)) - 2; \\ _Michel Marcus_, Nov 02 2015
%o A263878 (Magma) [(-1)^n*(Fibonacci(n-3) + n*Fibonacci(n-1)) - 2: n in [0..30]]; // _G. C. Greubel_, Jul 30 2018
%Y A263878 Cf. A000045, A014286.
%K A263878 sign,easy
%O A263878 0,4
%A A263878 _Vladimir Reshetnikov_, Oct 28 2015