A306471 Number of ways to write n as C(2w+1,2) + C(x+2,3) + C(y+3,4) + C(z+4,5) with w,x,y,z nonnegative integers, where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!).

1, 3, 3, 2, 4, 6, 5, 4, 4, 5, 7, 8, 6, 4, 5, 8, 8, 5, 4, 6, 7, 10, 10, 6, 6, 12, 13, 8, 7, 7, 6, 11, 9, 4, 3, 8, 16, 12, 8, 9, 9, 13, 14, 10, 7, 9, 18, 12, 6, 5, 4, 11, 10, 4, 2, 5, 19, 21, 11, 9, 13, 20, 16, 9, 6, 8, 17, 17, 4, 2, 9, 20, 17, 6, 9, 9, 15, 23, 14, 9, 15

Offset: 0

Author

Zhi-Wei Sun, Feb 17 2019

Keywords

• nonn

Comments

Conjecture 1: a(n) > 1 for all n > 0.

We have verified a(n) > 0 for all n = 0..5*10^6.

Conjecture 2: For each r = 0, 1, any positive integer can be written as w^2 + C(x,3) + C(y,4) + C(z,5), where w,x,y,z are nonnegative integers with w - r even.

See also A306462 and A306477 for similar conjectures.

Examples

			a(0) = 1 with 0 = C(1,2) + C(2,3) + C(3,4) + C(4,5).
a(3) = 2 with 3 = C(3,2) + C(2,3) + C(3,4) + C(4,5) = C(1,2) + C(3,3) + C(4,4) + C(5,5).
a(54) = 2 with 54 = C(3,2) + C(7,3) + C(6,4) + C(5,5) = C(3,2) + C(5,3) + C(7,4) + C(6,5).
a(69) = 1 with 69 = C(3,2) + C(5,3) + C(7,4) + C(7,5) = C(3,2) + C(5,3) + C(3,4) + C(8,5).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000217, A000290, A000292, A000332, A000389, A000797, A014105, A262813, A306460, A306462, A306477.

Programs

Mathematica

f[m_,n_]:=f[m,n]=Binomial[m+n-1,m];
HQ[n_]:=HQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==1;
tab={};Do[r=0;Do[If[f[5,z]>n,Goto[cc]];Do[If[f[4,y]>n-f[5,z],Goto[bb]];Do[If[f[3,x]>n-f[5,z]-f[4,y],Goto[aa]];If[HQ[n-f[5,z]-f[4,y]-f[3,x]],r=r+1],{x,0,n-f[5,z]-f[4,y]}];Label[aa],{y,0,n-f[5,z]}];Label[bb],{z,0,n}];Label[cc];tab=Append[tab,r],{n,0,80}];Print[tab]