A213043 Convolution of (1,-1,2,-2,3,-3,...) and A000045 (Fibonacci numbers).

1, 0, 3, 1, 7, 5, 16, 17, 38, 50, 94, 138, 239, 370, 617, 979, 1605, 2575, 4190, 6755, 10956, 17700, 28668, 46356, 75037, 121380, 196431, 317797, 514243, 832025, 1346284, 2178293, 3524594, 5702870, 9227482, 14930334, 24157835, 39088150, 63246005, 102334135

Offset: 0

Author

Clark Kimberling, Jun 10 2012

Keywords

• nonn
• easy

Comments

(1,-1,2,-2,3,-3,...) = ((-1)^n)*(1+floor(n/2)), which results from A001057 by removing its initial 0.

Examples

			a(5) = (1,-1,2,-2,3,-3)**(1,1,2,3,5,8)=1*8-1*5+2*3-2*2+3*1-3*1 = 5.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,1,-2,-1).

Crossrefs

Cf. A000045, A213500.

Programs

Mathematica

f[x_] := (1 - x^2) (1 + x); g[x] := 1 - x - x^2;
s = Normal[Series[1/(f[x] g[x]), {x, 0, 60}]]
c = CoefficientList[s, x]  (* A213043 *)
LinearRecurrence[{0, 3, 1, -2, -1}, {1, 0, 3, 1, 7}, 60]
Table[Fibonacci[n+1] + ((-1)^n (2n+1) - 1)/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

PARI

Vec(1/((1-x)*(1+x)^2*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Mar 16 2016

Formula

a(n) = 3*a(n-2)+a(n-3)-2*a(n-4)-a(n-5).

G.f.: 1/((1 + x)^2 * (1 - 2*x + x^3)).

From Vladimir Reshetnikov, Oct 29 2015: (Start)

a(n) = Fibonacci(n+1) + ((-1)^n*(2*n+1)-1)/4, where Fibonacci(n) = A000045(n).

Recurrence (4-term): a(0) = 1, a(1) = 0, a(2) = 3, (2*n+1)*a(n) = n + 1 - 2*a(n-1) + 4*(n+1)*a(n-2) + (2*n+3)*a(n-3).

(End)

From Colin Barker, Mar 16 2016: (Start)

a(n) = (-5-5*(-1)^n+2^(1-n)*sqrt(5)*(-(1-sqrt(5))^(1+n)+(1+sqrt(5))^(1+n))+10*(-1)^n*(1+n))/20.

a(n) = (sqrt(5)*2^(1-n)*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1))+10*(n+1)-10)/20 for n even.

a(n) = (sqrt(5)*2^(1-n)*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1))-10*(n+1))/20 for n odd.

(End)