A104581 Expansion of g.f. 1/(1 + x + x^3 + x^4).

1, -1, 1, -2, 2, -2, 3, -3, 3, -4, 4, -4, 5, -5, 5, -6, 6, -6, 7, -7, 7, -8, 8, -8, 9, -9, 9, -10, 10, -10, 11, -11, 11, -12, 12, -12, 13, -13, 13, -14, 14, -14, 15, -15, 15, -16, 16, -16, 17, -17, 17, -18, 18, -18, 19, -19, 19, -20, 20, -20, 21, -21, 21, -22, 22, -22, 23, -23, 23, -24, 24, -24, 25, -25, 25, -26, 26, -26

Offset: 0

Author

Paul Barry, Mar 16 2005

Keywords

• easy
• sign

Comments

Diagonal sums of Riordan array (1/(1+x+x^2+x^3+x^4),x/(1+x+x^2+x^3+x^4)).

Convolution of (n+1)(-1)^n and A010892.

Links

• Index entries for linear recurrences with constant coefficients, signature (-1,0,-1,-1).

Crossrefs

Cf. A008620, A010892.

Programs

Mathematica

CoefficientList[ Series[1/(1 + x + x^3 + x^4), {x, 0, 80}], x] (* Robert G. Wilson v, Mar 24 2005 *)
LinearRecurrence[{-1,0,-1,-1},{1,-1,1,-2},90] (* Harvey P. Dale, Jan 18 2019 *)

Formula

a(n) = floor((n + 3)/3)*(-1)^n.

a(n) = Sum_{k=0..n} ((n - k + 1)(-1)^(n-k)*2sin(Pi*k/3 + Pi/3)/sqrt(3)).

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

E.g.f.: exp(-x)*(6 - 3*x + exp(3*x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Feb 12 2023

Sum_{n>=0} 1/a(n) = log(2). - Amiram Eldar, Feb 14 2023