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A306571
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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)!).
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1
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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, 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, 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
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OFFSET
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0,2
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COMMENTS
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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.
We have verified a(n) > 0 for all n = 0..2*10^7.
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.
See also A306477 for a similar conjecture.
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LINKS
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EXAMPLE
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a(0) = 1 with 0 = 0*1 + C(3,4) + C(5,6) + C(7,8).
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).
a(220544) = 1 with 220544 = 151*152 + C(48,4) + C(14,6) + C(9,8).
a(809165) = 1 with 809165 = 295*296 + C(63,4) + C(10,6) + C(20,8).
a(16451641) = 1 with 16451641 = 2256*2257 + C(130,4) + C(12,6) + C(10,8).
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MATHEMATICA
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f[m_, n_]:=f[m, n]=Binomial[m+n-1, m];
TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];
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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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