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A299794
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Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x >= y >= 0 <= z <= w such that x or 2*y is a power of 4 (including 4^0 = 1) and x + 15*y is also a power of 4.
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30
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1, 1, 1, 1, 1, 3, 1, 1, 4, 2, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 6, 3, 1, 3, 3, 2, 2, 1, 3, 4, 2, 1, 5, 4, 4, 5, 1, 2, 3, 2, 5, 5, 2, 2, 8, 2, 2, 1, 5, 2, 4, 3, 4, 4, 4, 3, 6, 3, 2, 3, 3, 4, 3, 1, 3, 6, 4, 3, 11, 2, 2, 2, 4, 5, 1, 2, 3, 5, 3, 1
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OFFSET
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1,6
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COMMENTS
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Conjecture 1: a(n) > 0 for all n > 0. Also, for any integer n > 1 we can write n^2 as x^2 + y^2 + z^2 + w^2 with x >= y >= 0 <= z <= w such that 2*x or y is a power of 4 and also x + 15*y = 2^(2k+1) for some k = 0,1,2,....
Conjecture 2: Let d be 2 or 8, and let r be 0 or 1. Then any positive square n^2 can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x or y is a power of 2 and x + d*y = 2^(2k+r) for some k = 0,1,2,....
We have verified Conjecture 1 for n up to 10^7.
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LINKS
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EXAMPLE
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a(2) = 1 since 2^2 = 1^2 + 1^2 + 1^2 + 1^2 with 1 = 4^0 and 1 + 15*1 = 4^2.
a(5) = 1 since 5^2 = 4^2 + 0^2 + 0^2 + 3^2 with 4 = 4^1 and 4 + 15*0 = 4^1.
a(19) = 1 since 19^2 = 1^2 + 0^2 + 6^2 + 18^2 with 1 = 4^0 and 1 + 15*0 = 4^0.
a(159) = 1 since 159^2 = 34^2 + 2^2 + 75^2 + 136^2 with 2*2 = 4^1 and 34 + 15*2 = 4^3.
a(1998) = 1 since 1998^2 = 256^2 + 256^2 + 286^2 + 1944^2 with 256 = 4^4 and 256 + 15*256 = 4^6.
a(3742) = 1 since 3742^2 = 2176^2 + 128^2 + 98^2 + 3040^2 with 2*128 = 4^4 and 2176 + 15*128 = 4^6.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Pow[n_]:=Pow[n]=IntegerQ[Log[4, n]];
tab={}; Do[r=0; Do[If[Pow[2y]||Pow[4^k-15y], Do[If[SQ[n^2-y^2-(4^k-15y)^2-z^2], r=r+1], {z, 0, Sqrt[Max[0, (n^2-y^2-(4^k-15y)^2)/2]]}]],
{k, 0, Log[4, Sqrt[226]*n]}, {y, 0, Min[n, 4^(k-2)]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]
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CROSSREFS
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Cf. A000118, A000290, A000302, A271518, A281976, A299537, A299924, A300219, A300356, A300360, A300362, A300396.
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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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