%I #10 Feb 24 2019 13:04:54
%S 1,3,4,4,3,2,3,6,7,5,4,4,6,7,7,6,6,6,6,5,4,8,9,6,3,4,4,6,7,7,6,8,5,5,
%T 4,8,10,9,5,5,5,5,8,10,10,9,8,7,9,9,7,8,8,7,7,6,7,12,12,8,2,3,6,11,9,
%U 8,9,7,2,4,5,8,13,14,8,6,6,8,9,9,11,8,7,7,10,9,10,11,8,7,9,11,13,11,8,5,6,7,10,10,13
%N Number of ways to write n as w*(w+1) + C(x+3,4) + C(y+5,6) + C(z+7,8) with w,x,y,z nonnegative integers, where C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).
%C Conjecture: a(n) > 0 for all n >= 0. In other words, each n = 0,1,2,... can be written as 2*C(w,2) + C(x,4) + C(y,6) + C(z,8) with w,x,y,z positive integers.
%C We have verified a(n) > 0 for all n = 0..2*10^7.
%C Note that 10413917 is the least positive integer not representable as w^2 + C(x,4) + C(y,6) + C(z,8) with w,x,y,z nonnegative integers.
%C See also A306477 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A306571/b306571.txt">Table of n, a(n) for n = 0..10000</a>
%e a(0) = 1 with 0 = 0*1 + C(3,4) + C(5,6) + C(7,8).
%e a(60) = 2 with 60 = 0*1 + C(6,4) + C(5,6) + C(10,8) = 5*6 + C(4,4) + C(8,6) + C(8,8).
%e a(220544) = 1 with 220544 = 151*152 + C(48,4) + C(14,6) + C(9,8).
%e a(809165) = 1 with 809165 = 295*296 + C(63,4) + C(10,6) + C(20,8).
%e a(16451641) = 1 with 16451641 = 2256*2257 + C(130,4) + C(12,6) + C(10,8).
%t f[m_,n_]:=f[m,n]=Binomial[m+n-1,m];
%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];
%t tab={};Do[r=0;Do[If[f[8,z]>n,Goto[cc]];Do[If[f[6,y]>n-f[8,z],Goto[bb]];Do[If[f[4,x]>n-f[8,z]-f[6,y],Goto[aa]];If[TQ[n-f[8,z]-f[6,y]-f[4,x]],r=r+1],{x,0,n-f[8,z]-f[6,y]}];Label[aa],{y,0,n-f[8,z]}];Label[bb],{z,0,n}];Label[cc];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y Cf. A000332, A000579, A000581, A002378, A306459, A306460, A306462, A306471, A306477.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Feb 24 2019