A306462 Number of ways to write n as C(2w,2) + C(x+2,3) + C(y+3,4) + C(z+4,5), where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!), w is a positive integer and x,y,z are nonnegative integers.

1, 3, 3, 1, 1, 4, 7, 6, 2, 2, 6, 8, 5, 1, 2, 9, 11, 5, 1, 4, 9, 12, 7, 2, 4, 10, 12, 7, 4, 6, 10, 11, 6, 5, 5, 10, 15, 8, 4, 7, 11, 14, 9, 4, 5, 11, 14, 6, 6, 10, 15, 12, 5, 7, 8, 11, 14, 7, 5, 6, 11, 14, 12, 11, 6, 11, 15, 12, 7, 9, 18, 21, 12, 5, 5, 15, 19, 11, 3, 5

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 4, 5, 14, 19.

We have verified a(n) > 0 for all n = 1..5*10^6.

See also A306471 and A306477 for similar conjectures.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(1) = 1 with 1 = C(2,2) + C(2,3) + C(3,4) + C(4,5).
a(4) = 1 with 4 = C(2,2) + C(3,3) + C(4,4) + C(5,5).
a(5) = 1 with 5 = C(2,2) + C(4,3) + C(3,4) + C(4,5).
a(14) = 1 with 14 = C(4,2) + C(3,3) + C(4,4) + C(6,5).
a(19) = 1 with 19 = C(6,2) + C(4,3) + C(3,4) + C(4,5).

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]==3;
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-1-f[5, z]-f[4, y]}]; Label[aa], {y, 0, n-1-f[5, z]}]; Label[bb], {z, 0, n-1}]; Label[cc]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

Crossrefs

Cf. A000217, A000292, A000332, A000384, A000389, A000797, A262813, A306460, A306471, A306477.

Sequence in context: A133333 A296523 A171876 * A133332 A362895 A179680

Adjacent sequences: A306459 A306460 A306461 * A306463 A306464 A306465

Keyword

nonn

Author

Zhi-Wei Sun, Feb 17 2019

Status

approved