A166514 Zig-zag function for first two columns of a matrix (take numbers in consecutive pairs).

0, 1, 1, 0, 2, 1, 3, 0, 4, 1, 5, 0, 6, 1, 7, 0, 8, 1, 9, 0, 10, 1, 11, 0, 12, 1, 13, 0, 14, 1, 15, 0, 16, 1, 17, 0, 18, 1, 19, 0, 20, 1, 21, 0, 22, 1, 23, 0, 24, 1, 25, 0, 26, 1, 27, 0, 28, 1, 29, 0, 30, 1, 31, 0, 32, 1, 33, 0, 34, 1, 35, 0, 36, 1, 37, 0, 38, 1, 39, 0, 40, 1, 41, 0, 42, 1, 43, 0

Offset: 0

Author

Paul Barry, Oct 16 2009

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Programs

Magma

A166514:= func< n | ((n mod 2)*(1-(-1)^Floor((n+1)/2)) +((n+1) mod 2)*n)/2 >;
[A166514(n): n in [0..100]]; // G. C. Greubel, Aug 03 2024

Mathematica

CoefficientList[Series[(x + x^2 - x^3 + x^4)/((1 + x)^2*(1 - x)^2*(1 + x^2)), {x, 0, 50}], x] (* G. C. Greubel, May 15 2016 *)

SageMath

def A166514(n): return ((n%2)*(1+(-1)^((n-1)//2)) +((n+1)%2)*n)/2
[A166514(n) for n in range(101)] # G. C. Greubel, Aug 03 2024

Formula

G.f.: x*(1+x-x^2+x^3)/((1+x)^2*(1-x)^2*(1+x^2)) = x*(1+x-x^2+x^3)/(1-x^2-x^4+x^6).

a(n) = sin(Pi*n/2)/2 + (n-1)*(1 + (-1)^n)/4.

E.g.f.: (1/2)*(sin(x) + (1 + x)*sinh(x)). - G. C. Greubel, Aug 03 2024