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

0,3

Comments

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

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).

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)

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000045, A213500.

Sequence in context: A375584 A101624 A166519 * A319740 A275662 A110441

Adjacent sequences: A213040 A213041 A213042 * A213044 A213045 A213046

Keyword

nonn,easy

Author

Clark Kimberling, Jun 10 2012

Status

approved