A188134 - OEIS

%I #37 Oct 07 2023 05:06:33

%S 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,

%T 26,108,7,116,30,124,8,132,34,140,9,148,38,156,10,164,42,172,11,180,

%U 46,188,12,196,50,204,13,212,54,220,14,228,58,236,15,244,62

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

%H G. C. Greubel, <a href="/A188134/b188134.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2,0,0,0,-1).

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

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

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

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

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

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

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

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

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

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

%F From _Bruno Berselli_, Mar 22 2011: (Start)

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

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

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

%F a(n) = 8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)). - _Wesley Ivan Hurt_, Jul 06 2016

%F a(n) = lcm(4,n)/gcd(4,n). - _R. J. Mathar_, Feb 12 2019

%F Sum_{k=1..n} a(k) ~ (37/32)*n^2. - _Amiram Eldar_, Oct 07 2023

%p 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

%t Table[8 n/(11 + 9 Cos[Pi*n] + 12 Cos[n*Pi/2]), {n, 0, 80}] (* _Wesley Ivan Hurt_, Jul 06 2016 *)

%t 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 *)

%t LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,2,12,1,20,6,28},70] (* _Harvey P. Dale_, Aug 14 2019 *)

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

%o (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

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

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Mar 21 2011