A087620 #{0<=k<=n: k*n is divisible by 4}.

1, 1, 2, 1, 5, 2, 4, 2, 9, 3, 6, 3, 13, 4, 8, 4, 17, 5, 10, 5, 21, 6, 12, 6, 25, 7, 14, 7, 29, 8, 16, 8, 33, 9, 18, 9, 37, 10, 20, 10, 41, 11, 22, 11, 45, 12, 24, 12, 49, 13, 26, 13, 53, 14, 28, 14, 57, 15, 30, 15, 61, 16, 32, 16, 65, 17, 34, 17, 69, 18, 36, 18, 73, 19, 38, 19, 77, 20

Offset: 0

Author

Paul Barry, Sep 13 2003

Keywords

• easy
• nonn

Comments

With the similar remainder 1, 2 and 3 sequences provides a four-fold partition of A000027.

Links

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

Crossrefs

Cf. A026741, A077814, A087507.

Programs

Magma

I:=[1,1,2,1,5,2,4,2]; [n le 8 select I[n] else 2*Self(n-4)-Self(n-8): n in [1..80]]; // Vincenzo Librandi, May 03 2015

Mathematica

CoefficientList[Series[(3 x^4 + x^3 + 2 x^2 + x + 1)/((x - 1)^2 (x + 1)^2 (x^2 + 1)^2), {x, 0, 80}], x] (* Vincenzo Librandi, May 03 2015 *)

PARI

Vec((3*x^4+x^3+2*x^2+x+1)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100)) \\ Colin Barker, May 03 2015

Formula

a(n) = Sum_{k=0..n} if (k*n mod 4 = 0, 1, 0).

From Colin Barker, May 03 2015: (Start)

a(n) = (6+4*n+i^n*(-i+n)+(-i)^n*(i+n)+2*(-1)^n*(1+n))/8 where i=sqrt(-1).

a(n) = 2*a(n-4)-a(n-8) for n>7.

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

(End)