A157536 - OEIS

%I #15 Aug 28 2022 08:23:24

%S 1,3,6,110,1095,7161,37919,182367,860400,3893395,17126571,73368945,

%T 300879670,1173812745,4380376770,15753693858,54909242667,186135465987,

%U 614751720490,1979351435670,6214975736739,19042979887371,57000797449509

%N Half the number of length n integer sequences with sum zero and sum of squares 32.

%H Robert Israel and R. H. Hardin, <a href="/A157536/b157536.txt">Table of n, a(n) for n = 2..10000</a> (n = 2..50 from R. H. Hardin)

%F [cache enabling] count(n,s,ss)->count(n,t,tt) where t=s mod n, q=(t-s)/n, tt=ss+2*q*s+n*q^2; count(n,t,tt)=Sum_{i^2<=tt} count(n-1,t-i,tt-i^2). a(n)=count(n,0,32)/2.

%F G.f.: (1 - 30*x + 435*x^2 - 3960*x^3 + 25185*x^4 - 118206*x^5 + 420686*x^6 - 1134432*x^7 + 2206965*x^8 - 2665333*x^9 + 3050124*x^10 - 34121034*x^11 + 301563025*x^12 - 1647844221*x^13 + 6443298873*x^14 - 19087013776*x^15 + 43441736331*x^16 - 74712136848*x^17 + 90999467153*x^18 - 61749383676*x^19 - 17226181401*x^20 + 95716380496*x^21 - 106118533542*x^22 + 46133383200*x^23 + 12059750667*x^24 - 19161162021*x^25 + 2427632556*x^26 + 2280934382*x^27 + 225810123*x^28 + 5162259*x^29 + 16214*x^30) * x^2/(1-x)^33. - _Robert Israel_, Dec 25 2016

%p g:= proc(n,s,ss) option remember;

%p    if n = 1 then if ss = s^2 then return 1 else return 0 fi fi;

%p    procname(n-1,s,ss) +  add(procname(n-1,s-t, ss-t^2)

%p       +procname(n-1,s+t,ss-t^2),t=1..floor(sqrt(ss)));

%p end proc:

%p seq(g(n, 0, 32)/2, n=2..50); # _Robert Israel_, Dec 25 2016

%t g[n_, s_, ss_] := g[n, s, ss] = If[n == 1, If[ss == s^2, 1, 0], g[n-1, s, ss] + Sum[g[n-1, s-t, ss-t^2] + g[n-1, s+t, ss-t^2], {t, 1, Floor[Sqrt[ss]]}]];

%t Table[g[n, 0, 32]/2, {n, 2, 50}] (* _Jean-François Alcover_, Aug 28 2022, after _Robert Israel_ *)

%K nonn

%O 2,2

%A _R. H. Hardin_, Mar 02 2009