OFFSET
0,2
LINKS
Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
a(n) = 2*A083219(n).
a(n) = a(n-1) + 2*(n mod 2 + (n mod 4 -1)*(1- n mod 2)), a(0)=0.
a(n) = (3 - (-1)^n - (1+i)*(-i)^n - (1-i)*i^n + 2*n)/2 where i=sqrt(-1). - Colin Barker, Oct 13 2014
G.f.: -2*x*(x^3-x^2-x-1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Oct 13 2014
For n > 4, a(n) = a(n-4) + 4. - Zak Seidov, Feb 23 2017
G.f.: 1/(1-x)^2 + 1/(2*(1-x)) - 1/(2*(1+x)) - (1+x)/(1+x^2). - Michael Somos, Feb 23 2017
E.g.f.: (1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x). - Stefano Spezia, May 28 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 (A187832). - Amiram Eldar, Aug 21 2023
EXAMPLE
G.f. = 2*x + 4*x^2 + 6*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 10*x^7 + 8*x^8 + 10*x^9 + ...
MATHEMATICA
a[n_] := Mod[n, 4] + n; (* Michael Somos, Feb 23 2017 *)
PROG
(PARI) concat(0, Vec(-2*x*(x^3-x^2-x-1)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100))) \\ Colin Barker, Oct 13 2014
(PARI) {a(n) = n%4 + n}; /* Michael Somos, Feb 23 2017 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Apr 22 2003
STATUS
approved