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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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 A179680 A123562

Adjacent sequences:  A306459 A306460 A306461 * A306463 A306464 A306465

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 17 2019

STATUS

approved

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Last modified May 13 12:16 EDT 2021. Contains 343839 sequences. (Running on oeis4.)