A352636 Number of ways to write n as a^2 + 3*b^2 + c^4 + 2*d^4 + a*b*c*d, where a,b,c,d are nonnegative integers.

1, 2, 2, 3, 4, 3, 3, 2, 2, 2, 1, 1, 4, 3, 3, 2, 3, 3, 5, 4, 3, 3, 2, 3, 1, 2, 2, 3, 7, 6, 6, 1, 4, 3, 3, 3, 7, 5, 3, 6, 2, 3, 2, 3, 7, 5, 0, 1, 7, 6, 6, 7, 8, 5, 4, 1, 3, 4, 3, 3, 6, 2, 3, 6, 6, 5, 4, 4, 6, 5, 1, 2, 3, 3, 1, 6, 5, 6, 6, 2, 6, 6, 4, 5, 14, 6, 5, 2, 2, 2, 5, 3, 7, 8, 2, 5, 7, 6, 1, 6, 9

Offset

0,2

Comments

Conjecture: a(n) > 0 except for n = 46.

It seems that the only values of n > 500 with a(n) = 1 are 586, 616, 1097, 1208, 3496, 3736, 4486, 21736, 24568, 37870.

Links

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

Example

a(115) = 1 with 115 = 5^2 + 3*3^2 + 1^4 + 2*2^4 + 5*3*1*2.
a(1097) = 1 with 1097 = 29^2 + 3*0^2 + 4^4 + 2*0^4 + 29*0*4*0.
a(4468) = 1 with 4468 = 5^2 + 3*21^2 + 7^4 + 2*1^4 + 5*21*7*1.
a(21736) = 1 with 21736 = 50^2 + 3*65^2 + 9^4 + 2*0^4 + 50*65*9*0.
a(24568) = 1 with 24568 = 121^2 + 3*3^2 + 2^4 + 2*7^4 + 121*3*2*7.
a(37870) = 1 with 37870 = 31^2 + 3*27^2 + 5^4 + 2*7^4 + 31*27*5*7.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[4(n-2w^4-z^4-3*y^2)+(y*z*w)^2], r=r+1], {w, 0, (n/2)^(1/4)}, {z, 0, (n-2w^4)^(1/4)}, {y, 0, Sqrt[(n-2w^4-z^4)/3]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000583, A352627, A352628, A352629, A352632.

Sequence in context: A286543 A285732 A144909 * A117114 A307981 A262824

Adjacent sequences: A352633 A352634 A352635 * A352637 A352638 A352639

Keyword

nonn

Author

Zhi-Wei Sun, Mar 25 2022

Status

approved