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

0, -1, 1, -5, 7, -18, 30, -61, 107, -199, 351, -628, 1100, -1929, 3349, -5801, 9991, -17158, 29354, -50085, 85215, -144651, 244991, -414120, 698712, -1176913, 1979305, -3323981, 5574727, -9337914, 15623286, -26111053, 43594835, -72716239, 121181919, -201779356

Offset

0,4

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Eric Weisstein's MathWorld, Fibonacci Number.

Index entries for linear recurrences with constant coefficients, signature (-1,3,1,-3,1).

Formula

a(n) = (-1)^n*(F(n-3) + n*F(n-1)) - 2, where F(n) = A000045(n).

G.f.: x*(x^2+1)/((x-1)*(x^2-x-1)^2).

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.

Recurrences:

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).

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.

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).

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).

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

Example

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 + ...

Mathematica

Table[Sum[(-1)^k k Fibonacci[k], {k, 0, n}], {n, 0, 20}]
Table[(-1)^n (Fibonacci[n-3] + n Fibonacci[n-1]) - 2, {n, 0, 20}]

Prog

(PARI) concat(0, Vec(x*(x^2+1)/((x-1)*(x^2-x-1)^2) + O(x^40))) \\ Colin Barker, Oct 31 2015
(PARI) a(n) = (-1)^n*(fibonacci(n-3) + n*fibonacci(n-1)) - 2; \\ Michel Marcus, Nov 02 2015
(Magma) [(-1)^n*(Fibonacci(n-3) + n*Fibonacci(n-1)) - 2: n in [0..30]]; // G. C. Greubel, Jul 30 2018

Crossrefs

Cf. A000045, A014286.

Sequence in context: A214414 A153192 A370775 * A297937 A337349 A028319

Adjacent sequences: A263875 A263876 A263877 * A263879 A263880 A263881

Keyword

sign,easy

Author

Vladimir Reshetnikov, Oct 28 2015

Status

approved