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A284343
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Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y <= z such that 2*x + y - z is either zero or a power of 8 (including 8^0 = 1).
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1
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1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 1, 3, 2, 1, 3, 3, 2, 3, 5, 2, 3, 4, 6, 1, 3, 5, 1, 6, 1, 3, 7, 2, 2, 5, 6, 5, 6, 3, 6, 4, 1, 3, 4, 5, 4, 5, 7, 2, 3, 8, 6, 7, 3, 4, 8, 3, 2, 6, 3, 5, 7, 3, 8, 7, 2, 4, 10, 4, 4, 7, 9, 7, 2, 4, 2, 7, 3, 5, 11, 2, 4
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OFFSET
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0,3
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COMMENTS
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Conjecture: (i) For any c = 1,2,4, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y <= z such that c*(2*x+y-z) is either zero or a power of eight (including 8^0 = 1).
(ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that P(x,y,z,w) is either zero or a power of four (including 4^0 = 1), whenever P(x,y,z,w) is among the polynomials 2*x-y, x+y-z, x-y-z, x+y-2*z, 2*x+y-z, 2*x-y-z, 2*x-2*y-z, x+2*y-3*z, 2*x+2*y-2*z, 2*x+2*y-4*z, 3*x-2*y-z, x+3*y-3*z, 2*x+3*y-3*z, 4*x+2*y-2*z, 8*x+2*y-2*z, 2*(x-y)+z-w, 4*(x-y)+2*(z-w).
Part (i) of the conjecture is stronger than the first part of Conjecture 4.4 in the linked JNT paper (see also A273432).
Modifying the proofs of Theorem 1.1 and Theorem 1.2(i) in the linked JNT paper slightly, we see that for any a = 1,4 and m = 4,5,6 we can write each n = 0,1,2,... as a*x^m + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x is either zero or a power of two (including 2^0 = 1), and that for any b = 1,2 each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that b*(x-y) is either zero or a power of 4 (including 4^0 = 1).
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LINKS
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Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.
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EXAMPLE
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a(4) = 1 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 = 0 and 2*0 + 0 - 0 = 0.
a(5) = 1 since 5 = 1^2 + 0^2 + 2^2 + 0^2 with 0 < 2 and 2*1 + 0 - 2 = 0.
a(7) = 1 since 7 = 1^2 + 1^2 + 2^2 + 1^2 with 1 < 2 and 2*1 + 1 - 2 = 8^0.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 2 = 2 and 2*4 + 2 - 2 = 8.
a(138) = 1 since 138 = 3^2 + 5^2 + 10^2 + 2^2 with 5 < 10 and 2*3 + 5 - 10 = 8^0.
a(1832) = 1 since 1832 = 4^2 + 30^2 + 30^2 + 4^2 with 30 = 30 and 2*4 + 30 - 30 = 8.
a(2976) = 1 since 2976 = 20^2 + 16^2 + 48^2 + 4^2 with 16 < 48 and 2*20 + 16 - 48 = 8.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Pow[n_]:=Pow[n]=n==0||(n>0&&IntegerQ[Log[8, n]]);
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&Pow[2x+y-z], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[(n-x^2)/2]}, {z, y, Sqrt[n-x^2-y^2]}]; Print[n, " ", r], {n, 0, 80}]
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CROSSREFS
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Cf. A000079, A000118, A000290, A000302, A000578, A001018, A271518, A273432, A279612.
Sequence in context: A157654 A078692 A273432 * A033151 A046079 A319700
Adjacent sequences: A284340 A284341 A284342 * A284344 A284345 A284346
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun, Mar 25 2017
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STATUS
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approved
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