A225948 - OEIS

%I #80 Sep 08 2022 08:46:05

%S -1,-15,-3,-7,0,9,5,33,3,65,21,105,2,153,45,209,15,273,77,345,6,425,

%T 117,513,35,609,165,713,12,825,221,945,63,1073,285,1209,20,1353,357,

%U 1505,99,1665,437,1833,30,2009,525,2193,143

%N a(0) = -1; for n>0, a(n) = numerator(1/4 - 4/n^2).

%C Denominators are in A226008.

%C Fractions in lowest terms for n>0: -15/4, -3/4, -7/36, 0/1, 9/100, 5/36, 33/196, 3/16, 65/324, 21/100, 105/484, 2/9, 153/676, 45/196, 209/900, 15/64,...

%C If t(n) is the sequence with period 8: 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, ... (see A226044), then A226008(n) = 4*a(n) + t(n).

%H G. C. Greubel, <a href="/A225948/b225948.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = 3*a(n-8) -3*a(n-16) +a(n-24).

%F a(2n) = A061037(n), a(2n+1) = A145923(n-2) for A145923(-2)=-15, A145923(-1)=-7.

%F a(4n)   = A142705(n) for A142705(0)=-1, a(8n) = A000466(n);

%F a(4n+1) = A028566(4n-3) for A028566(-3)=-15;

%F a(4n+2) = A078371(n-1) for A078371(-1)=-3;

%F a(4n+3) = A028566(4n-1) for A028566(-1)=-7.

%F a(n+4)  = A106609(n) * A106609(n+8).

%F G.f.: -(1 +15*x +3*x^2 +7*x^3 -9*x^5 -5*x^6 -33*x^7 -6*x^8 -110*x^9 -30*x^10 -126*x^11 -2*x^12 -126*x^13 -30*x^14 -110*x^15 -3*x^16 -33*x^17 -5*x^18 -9*x^19 +7*x^21 +3*x^22 +15*x^23)/(1-x^8)^3. - _Bruno Berselli_, May 22 2013

%F a(n) = (n^2-16)*(6*cos(Pi*n/4)-54*cos(Pi*n/2)+6*cos(3*Pi*n/4)-219*(-1)^n+293)/512. - _Bruno Berselli_, May 22 2013

%F a(n+10) = a(n+2)*(n+14)/(n-2) for n=0,1 and n>2. - _Bruno Berselli_, May 22 2013

%t Join[{-1}, Table[Numerator[1/4 - 4/n^2], {n, 50}]] (* _Bruno Berselli_, May 24 2013 *)

%o (Magma) [-1] cat [Numerator(1/4-4/n^2): n in [1..50]]; // _Bruno Berselli_, May 22 2013

%o (PARI) concat([-1], vector(100, n, numerator(1/4 - 4/n^2))) \\ _G. C. Greubel_, Sep 19 2018

%Y Cf. A000466, A028566, A061037, A061041, A078371, A106609, A142705, A145923, A226008.

%K sign,frac,easy

%O 0,2

%A _Paul Curtz_, May 21 2013

%E Edited by _Bruno Berselli_, May 22 2013