A059816 - OEIS

%I #22 Oct 11 2022 21:42:28

%S 1,1,9,9,25,49,225,4,9,5,11,6,13,7,15,8,17,9,19,10,21,11,23,12,25,13,

%T 27,14,29,15,31,16,33,17,35,18,37,19,39,20,41,21,43,22,45,23,47,24,49,

%U 25,51,26,53,27,55,28,57,29,59,30,61,31,63,32,65,33,67,34

%N Let g_n be the ball packing n-width for the manifold torus X square; sequence gives denominator of (g_n/Pi)^2.

%H F. Miller Maley et al., <a href="https://projecteuclid.org/euclid.em/1045604678">Symplectic packings in cotangent bundles of tori</a>, Experimental Mathematics, 9 (No. 3, 2000), 435-455.

%F For n>=8, a(2n+1) = 2n+1, a(2n) = n. - _Ralf Stephan_, May 29 2004

%F From _Colin Barker_, Nov 06 2019: (Start)

%F G.f.: x*(1 + x + 7*x^2 + 7*x^3 + 8*x^4 + 32*x^5 + 184*x^6 - 85*x^7 - 416*x^8 + 46*x^9 + 218*x^10) / ((1 - x)^2*(1+x)^2).

%F a(n) = 2*a(n-2) - a(n-4) for n>=11.

%F a(n) = (1/4)*(3 - (-1)^n)*(n-1) for n>=8.

%F (End)

%F A059815(n) / a(n) = 2 / n, for n >= 8 [from Maley et al.]. - _Sean A. Irvine_, Oct 11 2022

%e 1, 1, 4/9, 4/9, 9/25, 16/49, 64/225, 1/4, ...

%Y Cf. A059812, A059813, A059814, A059815, A059817, A059818.

%K nonn,frac

%O 1,3

%A _N. J. A. Sloane_, Feb 24 2001

%E Edited by _N. J. A. Sloane_, May 23 2014

%E Duplicated a(8) removed and entry revised by _Sean A. Irvine_, Oct 11 2022