A188134 a(4*n) = n, a(1+2*n) = 4+8*n, a(2+4*n) = 2+4*n.

0, 4, 2, 12, 1, 20, 6, 28, 2, 36, 10, 44, 3, 52, 14, 60, 4, 68, 18, 76, 5, 84, 22, 92, 6, 100, 26, 108, 7, 116, 30, 124, 8, 132, 34, 140, 9, 148, 38, 156, 10, 164, 42, 172, 11, 180, 46, 188, 12, 196, 50, 204, 13, 212, 54, 220, 14, 228, 58, 236, 15, 244, 62

Offset: 0

Author

Paul Curtz, Mar 21 2011

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Crossrefs

Cf. A016825, A017113, A060819, A061037, A064680, A146160, A176895, A177499.

Programs

Magma

[(64-3*(1+(-1)^n)*(9+(-1)^(n div 2)))*n/16 : n in [0..80]]; // Wesley Ivan Hurt, Jul 06 2016

Maple

A188134:=n->8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)): seq(A188134(n), n=0..100); # Wesley Ivan Hurt, Jul 06 2016

Mathematica

Table[8 n/(11 + 9 Cos[Pi*n] + 12 Cos[n*Pi/2]), {n, 0, 80}] (* Wesley Ivan Hurt, Jul 06 2016 *)
CoefficientList[Series[x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2, {x, 0, 50}], x] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,2,12,1,20,6,28},70] (* Harvey P. Dale, Aug 14 2019 *)

PARI

x='x+O('x^50); concat([0], Vec(x*(4+2*x+12*x^2+x^3+12*x^4+ 2*x^5 +4*x^6)/(1-x^4)^2)) \\ G. C. Greubel, Sep 20 2018

Formula

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

a(n) = A176895(n) * A060819(n).

a(n) = (4*A061037(n+2))/(n+4).

a(n) = 4*n / A146160(n).

a(2*n) = A064680(n).

a(1+2*n) = A017113(n).

a(4*n) = a(-4+4*n) + 1.

a(1+4*n) = a(-3+4*n) + 16.

a(2+4*n) = a(-2+4*n) + 4.

a(3+4*n) = a(-1+4*n) + 16. See A177499.

From Bruno Berselli, Mar 22 2011: (Start)

G.f.: x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2.

a(n) = (64-3*(1+(-1)^n)*(9+i^n))*n/16 with i=sqrt(-1).

a(n)/a(n-4) = n/(n-4) for n>4. (End)

a(n) = 8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)). - Wesley Ivan Hurt, Jul 06 2016

a(n) = lcm(4,n)/gcd(4,n). - R. J. Mathar, Feb 12 2019

Sum_{k=1..n} a(k) ~ (37/32)*n^2. - Amiram Eldar, Oct 07 2023