A270073 -id:A270073 - cp's OEIS frontend

A272888 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w(x^2 + 8y^2 - z^2) a square, where w,x,y are nonnegative integers and z is a positive integer.

1, 2, 2, 1, 4, 5, 1, 2, 5, 5, 4, 4, 5, 8, 2, 2, 8, 6, 4, 6, 9, 5, 3, 4, 5, 12, 9, 1, 11, 8, 4, 2, 8, 9, 8, 7, 6, 12, 1, 5, 14, 10, 4, 8, 15, 9, 3, 4, 8, 14, 11, 5, 11, 16, 2, 6, 11, 6, 11, 4, 13, 13, 1, 1, 16, 17, 6, 9, 13, 9, 5, 7, 9, 19, 12, 6, 17, 8, 4, 6

Offset: 1

Zhi-Wei Sun, May 08 2016

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 7, 39, 63, 87, 5116, 2^(4k+2)*m (k = 0,1,2,... and m = 1, 7).

See arXiv:1604.06723 for more refinements of Lagrange's four-square theorem.

			a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 0*(0^2 + 8*0^2 - 1^2) = 0^2.
a(4) = 1 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 2 > 0 and 0*(0^2 + 8*0^2 - 2^2) = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and 2*(1^2 + 8*1^2 - 1^2) = 4^2.
a(28) = 1 since 28 = 2^2 + 2^2 + 4^2 + 2^2 with 2 > 0 and 2*(2^2 + 8*4^2 - 2^2) = 16^2.
a(39) = 1 since 39 = 1^2 + 3^2 + 2^2 + 5^2 with 5 > 0 and 1*(3^2 + 8*2^2 - 5^2) = 4^2.
a(63) = 1 since 63 = 2^2 + 5^2 + 3^2 + 5^2 with 5 > 0 and 2*(5^2 + 8*3^2 - 5^2) = 12^2.
a(87) = 1 since 87 = 2^2 + 1^2 + 9^2 + 1^2 with 1 > 0 and 2*(1^2 + 8*9^2 - 1^2) = 36^2.
a(5116) = 1 since 5116 = 65^2 + 9^2 + 9^2 + 27^2 with 27 > 0 and 65*(9^2 + 8*9^2 - 27^2) = 0^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620.

N:= 1000; # to get a(1)..a(N)
A:= Vector(N):
for z from 1 to floor(sqrt(N)) do
  for x from 0 to floor(sqrt(N-z^2)) do
    for y from 0 to floor(sqrt(N-z^2-x^2)) do
      q:= x^2 + 8*y^2 - z^2;
      if q < 0 then
        A[x^2+y^2+z^2]:= A[x^2+y^2+z^2]+1
      elif q = 0 then
        for w from 0 to floor(sqrt(N-z^2-x^2-y^2)) do
           m:= w^2 + x^2 + y^2 + z^2;
           A[m]:= A[m]+1;
        od
      else
        wm:= mul(`if`(t[2]::odd, t[1], 1), t=isqrfree(q)[2]);
        for j from 0 to floor((N-z^2-x^2-y^2)^(1/4)/sqrt(wm)) do
           m:= (wm*j^2)^2 + x^2 + y^2 + z^2;
           A[m]:= A[m]+1;
        od;
      fi
    od
  od
od:
convert(A,list); # Robert Israel, May 27 2016

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[Sqrt[n-x^2-y^2-z^2](x^2+8y^2-z^2)],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,1,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

Rick L. Shepherd, May 27 2016: I checked all the statements in each example.

A272977 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 3x^2y + z^2*w a square, where w is a nonzero integer and x,y,z are nonnegative integers with x >= z.

Original entry on oeis.org

2, 4, 1, 3, 8, 1, 1, 4, 3, 11, 3, 1, 9, 5, 3, 3, 10, 7, 6, 9, 3, 6, 1, 1, 11, 15, 4, 2, 13, 2, 2, 4, 4, 16, 5, 4, 13, 5, 2, 10, 12, 6, 5, 1, 12, 6, 1, 3, 7, 19, 2, 10, 10, 6, 3, 1, 2, 12, 7, 3, 15, 7, 4, 3, 16, 8, 6, 9, 5, 6, 1, 7, 12, 19, 3, 3, 7, 2, 4, 9

Offset: 1

Zhi-Wei Sun, May 11 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 7, 23, 47, 71, 147, 199, 263, 439, 16^k*m (k = 0,1,2,... and m = 6, 12, 24, 44, 56, 140, 156, 174, 204, 284, 4652).
(ii) For each ordered pair (a,b) = (7,1), (8,1), (9,2), any positive integer can be written as x^2 + y^2 + z^2 + w^2 with a*x^2*y + b*z^2*w a square, where x,y,z are nonnegative integers and w is a nonzero integer.
Compare this conjecture with the one in A270073.
See arXiv:1604.06723 for more refinements of Lagrange's four-square theorem.

			a(1) = 2 since 1 = 1^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 3*1^2*0 + 0^2*1 = 0^2, and also 1 = 1^2 + 0^2 + 0^2 + (-1)^2 with 1 > 0 and 3*1^2*0 + 0^2*(-1) = 0^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 1 = 1 and 3*1^2*0 + 1^2*1 = 1^2.
a(6) = 1 since 6 = 2^2 + 0^2 + 1^2 + 1^2 with 2 > 1 and 3*2^2*0 + 1^2*1 = 1^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + (-2)^2 with 1 = 1 and 3*1^2*1 + 1^2*(-2) = 1^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + (-3)^2 with 1 = 1 and 3*1^2*1 + 1*(-3) = 0^2.
a(23) = 1 since 23 = 3^2 + 1^2 + 3^2 + (-2)^2 with 3 = 3 and 3*3^2*1 + 3^2*(-2) = 3^2.
a(24) = 1 since 24 = 2^2 + 0^2 + 2^2 + 4^2 with 2 = 2 and 3*2^2*0 + 2^2*4 = 4^2.
a(44) = 1 since 44 = 3^2 + 5^2 + 3^2 + 1^2 with 3 = 3 and 3*3^2*5 + 3^2*1 = 12^2.
a(47) = 1 since 47 = 3^2 + 2^2 + 3^2 + (-5)^2 with 3 = 3 and 3*3^2*2 + 3^2*(-5) = 3^2.
a(56) = 1 since 56 = 6^2 + 0^2 + 2^2 + 4^2 with 6 > 2 and 3*6^2*0 + 2^2*4 = 4^2.
a(71) = 1 since 71 = 5^2 + 6^2 + 3^2 + (-1)^2 with 5 > 3 and 3*5^2*6 + 3^2*(-1) = 21^2.
a(140) = 1 since 140 = 5^2 + 3^2 + 5^2 + (-9)^2 with 5 = 5 and 3*5^2*3 + 5^2*(-9) = 0^2.
a(147) = 1 since 147 = 11^2 + 0^2 + 5^2 + 1^2 with 11 > 5 and 3*11^2*0 + 5^2*1 = 5^2.
a(156) = 1 since 156 = 7^2 + 3^2 + 7^2 + 7^2 with 7 = 7 and 3*7^2*3 + 7^2*7 = 28^2.
a(174) = 1 since 174 = 13^2 + 0^2 + 2^2 + 1^2 with 13 > 2 and 3*13^2*0 + 2^2*1 = 2^2.
a(199) = 1 since 199 = 9^2 + 1^2 + 9^2 + 6^2 with 9 = 9 and 3*9^2*1 + 9^2*6 = 27^2.
a(204) = 1 since 204 = 1^2 + 9^2 + 1^2 + (-11)^2 with 1 = 1 and 3*1^2*9 + 1^2*(-11) = 4^2.
a(263) = 1 since 263 = 3^2 + 14^2 + 3^2 + 7^2 with 3 = 3 and
3*3^2*14 + 3^2*7 = 21^2.
a(284) = 1 since 284 = 13^2 + 3^2 + 5^2 + (-9)^2 with 13 > 5 and 3*13^2*3 + 5^2*(-9) = 36^2.
a(439) = 1 since 439 = 13^2 + 5^2 + 7^2 + (-14)^2 with 13 > 7 and 3*13^2*5 + 7^2*(-14) = 43^2.
a(4652) = 1 since 4652 = 11^2 + 21^2 + 11^2 + (-63)^2 with 11 = 11 and 3*11^2*21 + 11^2*(-63) = 0^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[3x^2*y+z^2*(-1)^k*Sqrt[n-x^2-y^2-z^2]],r=r+1], {z,0,Sqrt[(n-1)/2]},{x,z,Sqrt[n-1-z^2]},{y,0,Sqrt[n-1-x^2-z^2]},{k,0,1}];Print[n, " ",r]; Continue, {n,1,80}]

A273021 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 2xy + yz - zw - w*x a square, where w is a positive integer and x,y,z are nonnegative integers with x <= y.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 1, 2, 2, 1, 3, 3, 4, 2, 2, 3, 5, 2, 2, 4, 1, 1, 3, 3, 4, 7, 4, 4, 1, 1, 1, 4, 4, 2, 4, 4, 6, 5, 2, 5, 7, 3, 3, 3, 4, 1, 3, 5, 4, 5, 6, 2, 8, 1, 4, 4, 4, 3, 2, 5, 5, 4, 2, 5, 7, 2, 3, 4, 5, 1, 5, 4, 5, 6, 5, 3, 4, 3, 2

Offset: 1

Zhi-Wei Sun, May 13 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 11, 31, 47, 55, 71, 105, 115, 119, 253, 383, 385, 4^k*m (k = 0,1,2,... and m = 2, 22, 23, 30, 330).
(ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with (x+y)*(z+w) a square, where w is an integer and x,y,z are nonnegative integers with x <= y >= z >= |w|.
See arXiv:1604.06723 for more conjectural refinements of Lagrange's four-square theorem.

			a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0 and 2*0*0 + 0*0 - 0*1 - 1*0 = 0^2.
a(2) = 1 since 2 = 0^2 + 1^2 + 0^2 + 1^2 with 0 < 1 and 2*0*1 + 1*0 - 0*1 - 1*0 = 0^2.
a(11) = 1 since 11 = 0^2 + 1^2 + 3^2 + 1^2 with 0 < 1 and 2*0*1 + 1*3 - 3*1 - 1*0 = 0^2.
a(22) = 1 since 22 = 0^2 + 3^2 + 2^2 + 3^2 with 0 < 3 and 2*0*3 + 3*2 - 2*3 - 3*0 = 0^2.
a(23) = 1 since 23 = 2^2 + 3^2 + 3^2 + 1^2 with 2 < 3 and 2*2*3 + 3*3 - 3*1 - 1*2 = 4^2.
a(30) = 1 since 30 = 1^2 + 3^2 + 2^2 + 4^2 with 1 < 3 and 2*1*3 + 3*2 - 2*4 - 4*1 = 0^2.
a(31) = 1 since 31 = 3^2 + 3^2 + 2^2 + 3^2 with 3 = 3 and
2*3*3 + 3*2 - 2*3 -3*3 = 3^2.
a(47) = 1 since 47 = 3^2 + 5^2 + 2^2 + 3^2 with 3 < 5 and 2*3*5 + 5*2 - 2*3 - 3*3 = 5^2.
a(55) = 1 since 55 = 1^2 + 7^2 + 2^2 + 1^2 with 1 < 7 and 2*1*7 + 7*2 - 2*1 - 1*1 = 5^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 < 5 and 2*1*5 + 5*3 - 3*6 - 6*1 = 1^2.
a(105) = 1 since 105 = 1^2 + 6^2 + 2^2 + 8^2 with 1 < 6 and 2*1*6 + 6*2 - 2*8 - 8*1 = 0^2.
a(115) = 1 since 115 = 1^2 + 8^2 + 7^2 + 1^2 with 1 < 8 and 2*1*8 + 8*7 - 7*1 - 1*1 = 8^2.
a(119) = 1 since 119 = 1^2 + 6^2 + 1^2 + 9^2 with 1 < 6 and 2*1*6 + 6*1 - 1*9 - 9*1 = 0^2.
a(253) = 1 since 253 = 2^2 + 8^2 + 11^2 + 8^2 with 2 < 8 and 2*2*8 + 8*11 - 11*8 - 8*2 = 4^2.
a(330) = 1 since 330 = 4^2 + 13^2 + 8^2 + 9^2 with 4 < 13 and 2*4*13 + 13*8 - 8*9 - 9*4 = 10^2.
a(383) = 1 since 383 = 9^2 + 14^2 + 5^2 + 9^2 with 9 < 14 and 2*9*14 + 14*5 - 5*9 - 9*9 = 14^2.
a(385) = 1 since 385 = 4^2 + 12^2 + 0^2 + 15^2 with 4 < 12 and 2*4*12 + 12*0 - 0*15 - 15*4 = 6^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[y*(2x+z)-Sqrt[n-x^2-y^2-z^2]*(x+z)],r=r+1],{x,0,Sqrt[(n-1)/2]},{y,x,Sqrt[n-1-x^2]},{z,0,Sqrt[n-1-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A273108 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x + y)^2 + (4z)^2 a square, where x,y,z,w are nonnegative integers with x <= y > z.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 1, 2, 3, 5, 3, 1, 3, 4, 1, 1, 5, 4, 3, 3, 3, 4, 1, 3, 5, 9, 4, 1, 6, 5, 3, 2, 5, 7, 6, 3, 3, 7, 1, 5, 9, 5, 3, 3, 6, 5, 1, 1, 6, 10, 6, 3, 6, 9, 3, 4, 4, 5, 8, 1, 6, 8, 2, 1, 10, 10, 2, 5, 6, 6, 2, 4, 6, 11, 7, 3, 6, 5, 2, 3

Offset: 1

Zhi-Wei Sun, May 15 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 39, 47, 95, 543, 4^k*m (k = 0,1,2,... and m = 1, 3, 7, 15, 23, 135, 183).
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that (a*x+b*y)^2 + (c*z)^2 is a square, whenever (a,b,c) is among the triples (1,2,4), (1,2,12), (1,4,8), (1,4,12), (1,10,20), (1,15,12), (2,7,20), (2,7,60), (2,21,60), (3,3,4), (3,3,40), (3,4,12), (3,5,60), (3,6,20), (3,9,20), (3,11,24), (3,12,8), (3,27,20), (3,27,56), (3,29,60), (3,30,28), (3,45,20), (4,4,3), (4,4,5), (4,4,9), (4,4,15), (4,8,5), (4,12,15), (4,12,21), (4,12,45), (4,16,45), (4,19,40), (4,20,21), (4,36,21), (4,36,33), (4,52,63), (5,5,8), (5,5,12), (5,5,24), (5,6,12), (5,8,24), (5,10,4), (5,15,24), (5,25,16), (5,30,12), (5,35,48), (5,40,24), (6,10,15), (6,15,28), (6,45,28), (7,7,20), (7,7,24), (7,21,12), (7,63,36), (8,8,15), (8,12,45), (8,16,35), (8,16,45), (8,32,15), (8,32,21), (8,48,45), (9,9,40), (9,18,28), (9,27,16), (9,45,20), (10,15,12), (10,25,28), (11,11,60), (12,12,5), (12,12,35), (12,20,63), (12,60,55).
See also A271714, A273107, A273110 and A273134 for similar conjectures related to Pythagorean triples. For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

			a(1) = 1 since 1 = 0^2 + 1^2 + 0^2 + 0^2 with 0 < 1 > 0 and (0+1)^2 + (4*0)^2 = 1^2.
a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1 = 1 > 0 and (1+1)^2 + (4*0)^2 = 2^2.
a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with 1 < 2 > 1 and (1+2)^2 + (4*1)^2 = 5^2.
a(15) = 1 since 15 = 1^2 + 2^2 + 1^2 + 3^2 with 1 < 2 > 1 and (1+2)^2 + (4*1)^2 = 5^2.
a(23) = 1 since 23 = 3^2 + 3^2 + 2^2 + 1^2 with 3 = 3 > 2 and (3+3)^2 + (4*2)^2 = 10^2.
a(39) = 1 since 39 = 1^2 + 5^2 + 2^2 + 3^2 with 1 < 5 > 2 and (1+5)^2 + (4*2)^2 = 10^2.
a(47) = 1 since 47 = 3^2 + 3^2 + 2^2 + 5^2 with 3 = 3 > 2 and (3+3)^2 + (4*2)^2 = 10^2.
a(95) = 1 since 95 = 3^2 + 7^2 + 6^2 + 1^2 with 3 < 7 > 6 and (3+7)^2 + (4*6)^2 = 26^2.
a(135) = 1 since 135 = 3^2 + 6^2 + 3^2 + 9^2 with 3 < 6 > 3 and (3+6)^2 + (4*3)^2 = 15^2.
a(183) = 1 since 183 = 2^2 + 7^2 + 3^2 + 11^2 with 2 < 7 > 3 and (2+7)^2 + (4*3)^2 = 15^2.
a(543) = 1 since 543 = 2^2 + 13^2 + 9^2 + 17^2 with 2 < 13 > 9 and (2+13)^2 + (4*9)^2 = 39^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273110, A273134.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+y)^2+16*z^2],r=r+1],{x,0,Sqrt[n/2]},{y,x,Sqrt[n-x^2]},{z,0,Min[y-1,Sqrt[n-x^2-y^2]]}];Print[n," ",r];Continue,{n,1,80}]

All statements in examples checked by Rick L. Shepherd, May 29 2016

A273107 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (8x+12y)^2 + (15*z)^2 a square, where x,y,z,w are nonnegative integers with x+y > 0 and z > 0.

Original entry on oeis.org

0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 3, 4, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 1, 1, 1, 1, 4, 3, 3, 4, 1, 1, 5, 3, 2, 3, 3, 5, 2, 1, 2, 1, 3, 3, 3, 3, 2, 4, 5, 5, 2, 4, 5, 6, 1, 3, 7, 3, 5, 4, 2, 6, 4, 1, 5, 4, 5, 4, 7, 7, 4, 3, 5, 4, 5, 6, 2, 10, 3, 1

Offset: 1

Zhi-Wei Sun, May 15 2016

nonn

Conjecture: a(n) > 0 for all n > 5, and a(n) = 1 only for n = 7, 9, 23, 25, 31, 55, 2^k*m (k = 1,2,... and m = 1, 5), 2^(2k+1)*m (k = 0,1,2,... and m = 3, 13, 21).
This conjecture implies that any integer n > 5 can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 8*x+12*y and 15*z are the two legs of a right triangle with positive integer sides.
See also A271714, A273108, A273110 and A273134 for similar conjectures related to Pythagorean triples. For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

			a(2) = 1 since 2 = 1^2 + 0^2 + 1^2 + 0^2 with 1 + 0 > 0 < 1 and (8*1+12*0)^2 + (15*1)^2 = 17^2.
a(4) = 1 since 4 = 1^2 + 1^2 + 1^2 + 1^2 with 1 + 1 > 0 < 1 and (8*1+12*1)^2 + (15*1)^2 = 25^2.
a(6) = 1 since 6 = 1^2 + 0^2 + 1^2 + 2^2 with 1 + 0 > 0 < 1 and (8*1+12*0)^2 + (15*1)^2 = 17^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 + 1 > 0 < 1 and (8*1+12*1)^2 + (15*1)^2 = 25^2.
a(9) = 1 since 9 = 2^2 + 0^2 + 2^2 + 1^2 with 2 + 0 > 0 < 2 and (8*2+12*0)^2 + (15*2)^2 = 34^2.
a(10) = 1 since 10 = 0^2 + 3^2 + 1^2 + 0^2 with 0 + 3 > 0 < 1 and (8*0+12*3)^2 + (15*1)^2 = 39^2.
a(20) = 1 since 20 = 3^2 + 1^2 + 1^2 + 3^2 with 3 + 1 > 0 < 1 and (8*3+12*1)^2 + (15*1)^2 = 39^2.
a(23) = 1 since 23 = 2^2 + 1^2 + 3^2 + 3^2 with 2 + 1 > 0 < 3 and (8*2+12*1)^2 + (15*3)^2 = 53^2.
a(25) = 1 since 25 = 1^2 + 2^2 + 4^2 + 2^2 with 1 + 2 > 0 < 4 and (8*1+12*2)^2 + (15*4)^2 = 68^2.
a(26) = 1 since 26 = 0^2 + 3^2 + 1^2 + 4^2 with 0 + 3 > 0 < 1 and (8*0+12*3)^2 + (15*1)^2 = 39^2.
a(31) = 1 since 31 = 3^2 + 3^2 + 3^2 + 2^2 with 3 + 3 > 0 < 3 and (8*3+12*3)^2 + (15*3)^2 = 75^2.
a(42) = 1 since 42 = 2^2 + 2^2 + 5^2 + 3^2 with 2 + 2 > 0 < 5 and (8*2+12*2)^2 + (15*5)^2 = 85^2.
a(55) = 1 since 55 = 6^2 + 1^2 + 3^2 + 3^2 with 6 + 1 > 0 < 3 and (8*6+12*1)^2 + (15*3)^2 = 75^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273108, A273110, A273134.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(8x+12y)^2+(15z)^2],r=r+1],{x,0,Sqrt[n-1]},{y,Max[0,1-x],Sqrt[n-1-x^2]},{z,1,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A273110 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+4y+4z)^2 + (9x+3y+3*z)^2 a square, where x,y,z,w are nonnegative integers with y > 0 and y >= z <= w.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 1, 2, 3, 3, 3, 2, 2, 2, 2, 1, 5, 6, 2, 2, 2, 3, 1, 3, 3, 4, 6, 1, 4, 4, 1, 2, 6, 5, 3, 3, 2, 5, 1, 3, 6, 5, 4, 3, 4, 3, 1, 2, 4, 7, 7, 2, 4, 8, 1, 2, 6, 3, 4, 2, 4, 5, 4, 1, 7, 8, 4, 5, 4, 4, 1, 6, 5, 7, 5, 2, 4, 5, 1, 2

Offset: 1

Zhi-Wei Sun, May 15 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m (k = 0,1,2,... and m = 1, 7, 23, 31, 39, 47, 55, 71, 79, 119, 151, 191, 311, 671).
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with (x+y+z)^2 + (4*(x+y-z))^2 a square, where x,y,z,w are nonnegative integers with x+y >= z.
(iii) For each tuple (a,b,c,d,e,f) = (1,1,1,3,6,-3), (1,1,1,4,12,-12), (1,1,2,1,1,-5), (1,1,2,1,8,-5), (1,1,2,3,3,-3), (1,1,2,4,4,-8), (1,3,11,12,4,4), (1,3,14,16,4,4), (1,3,14,18,4,2), (1,3,20,16,4,12), (1,4,11,6,3,3), (1,5,13,12,12,12), (1,5,14,15,12,21), (1,6,6,16,8,8), (1,6,14,12,8,8), (1,6,14,16,8,4), (1,6,17,20,8,4), (1,6,20,20,8,8), (1,7,8,4,2,6), (1,7,8,10,5,15), (1,7,9,10,5,12), (1,7,15,4,2,8), (1,7,15,10,5,20), any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that (a*x+b*y+c*z)^2 + (d*x+e*y+f*z)^2 is a square.
It was proved in arXiv:1604.06723 that any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y > 0 such that x+4*y+4*z and 9*x+3*y+3*z are the two legs of a right triangle with positive integer sides.
See also A271714, A273107, A273108 and A273134 for similar conjectures related to Pythagorean triples. For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

			a(1) = 1 since 1 = 0^2 + 1^2 + 0^2 + 0^2 with 1 > 0 = 0 and (0+4*1+4*0)^2 + (9*0+3*1+3*0)^2 = 5^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 0 < 1 = 1 = 1 and (2+4*1+4*1)^2 + (9*2+3*1+3*1)^2 = 26^2.
a(23) = 1 since 23 = 3^2 + 2^2 + 1^2 + 3^2 with 2 > 1 < 3 and (3+4*2+4*1)^2 + (9*3+3*2+3*1)^2 = 39^2.
a(31) = 1 since 31 = 2^2 + 1^2 + 1^2 + 5^2 with 0 < 1 = 1 < 5 and (2+4*1+4*1)^2 + (9*2+3*1+3*1)^2 = 26^2.
a(39) = 1 since 39 = 3^2 + 2^2 + 1^2 + 5^2 with 2 > 1 < 5 and (3+4*2+4*1)^2 + (9*3+3*2+3*1)^2 = 39^2.
a(47) = 1 since 47 = 5^2 + 3^2 + 2^2 + 3^2 with 3 > 2 < 3 and (5+4*3+4*2)^2 + (9*5+3*3+3*2)^2 = 65^2.
a(55) = 1 since 55 = 2^2 + 1^2 + 1^2 + 7^2 with 0 < 1 = 1 < 7 and (2+4*1+4*1)^2 + (9*2+3*1+3*1)^2 = 26^2.
a(71) = 1 since 71 = 6^2 + 5^2 + 1^2 + 3^2 with 5 > 1 < 3 and (6+4*5+4*1)^2 + (9*6+3*5+3*1)^2 = 78^2.
a(79) = 1 since 79 = 6^2 + 3^2 + 3^2 + 5^2 with 0 < 3 = 3 < 5 and (6+4*3+4*3)^2 + (9*6+3*3+3*3)^2 = 78^2.
a(119) = 1 since 119 = 5^2 + 3^2 + 2^2 + 9^2 with 3 > 2 < 9 and (5+4*3+4*2)^2 + (9*5+3*3+3*2)^2 = 65^2.
a(151) = 1 since 151 = 9^2 + 6^2 + 3^2 + 5^2 with 6 > 3 < 5 and (9+4*6+4*3)^2 + (9*9+3*6+3*3)^2 = 117^2.
a(191) = 1 since 191 = 10^2 + 9^2 + 1^2 + 3^2 with 9 > 1 < 3 and (10+4*9+4*1)^2 + (9*10+3*9+3*1)^2 = 130^2.
a(311) = 1 since 311 = 7^2 + 6^2 + 1^2 + 15^2 with 6 > 1 < 15 and (7+4*6+4*1)^2 + (9*7+3*6+3*1)^2 = 91^2.
a(671) = 1 since 671 = 17^2 + 11^2 + 6^2 + 15^2 with 11 > 6 < 15 and (17+4*11+4*6)^2 + (9*17+3*11+3*6)^2 = 221^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273134.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+4y+4z)^2+(9x+3y+3z)^2],r=r+1],{x,0,Sqrt[n]},{z,0,Sqrt[(n-x^2)/3]},{y,Max[1,z],Sqrt[n-x^2-2z^2]}];Print[n," ",r];Continue,{n,1,80}]

A273134 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+8y+8z+15w)^2+(6(x+y+z+w))^2 a square, where x,y,z,w are nonnegative integers with y < z.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 4, 2, 1, 1, 2, 1, 1, 3, 2, 3, 3, 1, 1, 2, 1, 3, 3, 1, 3, 3, 3, 1, 1, 2, 5, 3, 2, 3, 1, 2, 2, 3, 2, 2, 4, 2, 4, 3, 1, 3, 4, 2, 4, 3, 1, 3, 1, 2, 5, 4, 3, 2, 3, 1, 4, 5, 2, 3, 5, 3, 2, 2, 1

Offset: 1

Zhi-Wei Sun, May 16 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 7, 11, 15, 21, 23, 35, 39, 71, 95, 4^k*m (k = 0,1,2,... and m = 1, 2, 5, 6, 10, 14, 29, 30, 46, 62, 94, 110, 142, 190, 238, 334, 446).
(ii) For each polynomial P(x,y,z,w) = (x+3y+6z+17w)^2 + (20x+4y+8z+4w)^2, (x+3y+9z+17w)^2 + (20x+4y+12z+4w)^2, (x+3y+11z+15w)^2 + (12x+4y+4z+20w)^2, (3*(x+2y+3z+4w))^2 + (4*(x+4y+3z+2w))^2, (3*(x+2y+3z+4w))^2 + (4*(x+5y+3z+w))^2, any natural number 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 a square.
Part (i) of this conjecture implies that any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x+8*y+8*z+15*w and 6*(x+y+z+w) are the two legs of a right triangle with positive integer sides. If a nonnegative integer n is not of the form 4^k*(16m+14) (k,m = 0,1,2,...), then n can be written as w^2+x^2+y^2+z^2 with w = x and hence (x+8y+8z+15w)^2 + (6(x+y+z+w))^2 = (8(2x+y+z))^2 + (6(2x+y+z))^2 = (10(2x+y+z))^2. Similar remarks apply to part (ii) of the conjecture.
See also A271714, A273107, A273108 and A273110 for similar conjectures related to Pythagorean triples. For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

			a(1) = 1 since 1 = 0^2 + 0^2 + 1^2 + 0^2 with 0 < 1 and (0+8*0+8*1+15*0)^2 + (6*(0+0+1+0))^2 = 10^2.
a(2) = 1 since 2 = 1^2 + 0^2 + 1^2 + 0^2 with 0 < 1 and (1+8*0+8*1+15*0)^2 + (6*(1+0+1+0))^2 = 15^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 0 < 1 and (1+8*0+8*1+15*1)^2 + (6*(1+0+1+1))^2 = 30^2.
a(5) = 1 since 5 = 0^2 + 1^2 + 2^2 + 0^2 with 1 < 2 and (0+8*1+8*2+15*0)^2 + (6*(0+1+2+0))^2 = 30^2.
a(6) = 1 since 6 = 1^2 + 0^2 + 2^2 + 1^2 with 0 < 2 and (1+8*0+8*2+15*1)^2 + (6*(1+0+2+1))^2 = 40^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 2^2 + 1^2 with 1 < 2 and (1+8*1+8*2+15*1)^2 + (6*(1+1+2+1))^2 = 50^2.
a(10) = 1 since 10 = 0^2 + 1^2 + 3^2 + 0^2 with 1 < 3 and (0+8*1+8*3+15*0)^2 + (6*(0+1+3+0))^2 = 40^2.
a(11) = 1 since 11 = 1^2 + 0^2 + 3^2 + 1^2 with 0 < 3 and (1+8*0+8*3+15*1)^2 + (6*(1+0+3+1))^2 = 50^2.
a(14) = 1 since 14 = 3^2 + 1^2 + 2^2 + 0^2 with 1 < 2 and (3+8*1+8*2+15*0)^2 + (6*(3+1+2+0))^2 = 45^2.
a(15) = 1 since 15 = 1^2 + 2^2 + 3^2 + 1^2 with 2 < 3 and (1+8*2+8*3+15*1)^2 + (6*(1+2+3+1))^2 = 70^2.
a(21) = 1 since 21 = 2^2 + 2^2 + 3^2 + 2^2 with 2 < 3 and (2+8*2+8*3+15*2)^2 + (6*(2+2+3+2))^2 = 90^2.
a(23) = 1 since 23 = 3^2 + 1^2 + 2^2 + 3^2 with 1 < 2 and (3+8*1+8*2+15*3)^2 + (6*(3+1+2+3))^2 = 90^2.
a(29) = 1 since 29 = 0^2 + 2^2 + 5^2 + 0^2 with 2 < 5 and (0+8*2+8*5+15*0)^2 + (6*(0+2+5+0))^2 = 70^2.
a(30) = 1 since 30 = 5^2 + 0^2 + 2^2 + 1^2 with 0 < 2 and (5+8*0+8*2+15*1)^2 + (6*(5+0+2+1))^2 = 60^2.
a(35) = 1 since 35 = 3^2 + 1^2 + 4^2 + 3^2 with 1 < 4 and (3+8*1+8*4+15*3)^2 + (6*(3+1+4+3))^2 = 110^2.
a(39) = 1 since 39 = 1^2 + 1^2 + 6^2 + 1^2 with 1 < 6 and (1+8*1+8*6+15*1)^2 + (6*(1+1+6+1))^2 = 90^2.
a(46) = 1 since 46 = 6^2 + 0^2 + 3^2 + 1^2 with 0 < 3 and (6+8*0+8*3+15*1)^2 + (6*(6+0+3+1))^2 = 75^2.
a(62) = 1 since 62 = 6^2 + 1^2 + 5^2 + 0^2 with 1 < 5 and (6+8*1+8*5+15*0)^2 + (6*(6+1+5+0))^2 = 90^2.
a(71) = 1 since 71 = 3^2 + 2^2 + 7^2 + 3^2 with 2 < 7 and (3+8*2+8*7+15*3)^2 + (6*(3+2+7+3))^2 = 150^2.
a(94) = 1 since 94 = 9^2 + 0^2 + 3^2 + 2^2 with 0 < 3 and (9+8*0+8*3+15*2)^2 + (6*(9+0+3+2))^2 = 105^2.
a(95) = 1 since 95 = 5^2 + 3^2 + 6^2 + 5^2 with 3 < 6 and (5+8*3+8*6+15*5)^2 + (6*(5+3+6+5))^2 = 190^2.
a(110) = 1 since 110 = 10^2 + 0^2 + 1^2 + 3^2 with 0 < 1 and (10+8*0+8*1+15*3)^2 + (6*(10+0+1+3))^2 = 105^2.
a(142) = 1 since 142 = 11^2 + 1^2 + 4^2 + 2^2 with 1 < 4 and (11+8*1+8*4+15*2)^2 + (6*(11+1+4+2))^2 = 135^2.
a(190) = 1 since 190 = 12^2 + 3^2 + 6^2 + 1^2 with 3 < 6 and (12+8*3+8*6+15*1)^2 + (6*(12+3+6+1))^2 = 165^2.
a(238) = 1 since 238 = 13^2 + 2^2 + 8^2 + 1^2 with 2 < 8 and (13+8*2+8*8+15*1)^2 + (6*(13+2+8+1))^2 = 180^2.
a(334) = 1 since 334 = 4^2 + 2^2 + 5^2 + 17^2 with 2 < 5 and
(4+8*2+8*5+15*17)^2 + (6*(4+2+5+17))^2 = 357^2.
a(446) = 1 since 446 = 17^2 + 6^2 + 11^2 + 0^2 with 6 < 11 and (17+8*6+8*11+15*0)^2 + (6*(17+6+11+0))^2 = 255^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+8*y+8*z+15*Sqrt[n-x^2-y^2-z^2])^2+36(x+y+z+Sqrt[n-x^2-y^2-z^2])^2],r=r+1],{x,0,Sqrt[n-1]},{y,0,(Sqrt[2(n-x^2)-1]-1)/2},{z,y+1,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A273278 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^2y^2 + 3y^2z^2 + 2z^2*w^2 is a square.

Original entry on oeis.org

1, 4, 4, 2, 4, 8, 4, 1, 4, 8, 8, 4, 2, 9, 6, 3, 4, 12, 9, 6, 8, 8, 5, 2, 4, 14, 15, 6, 1, 14, 9, 2, 4, 9, 12, 8, 8, 9, 11, 1, 8, 18, 7, 4, 4, 17, 8, 3, 2, 12, 18, 9, 9, 17, 15, 4, 6, 8, 8, 10, 3, 15, 13, 5, 4, 22, 15, 6, 12, 15, 13

Offset: 0

Zhi-Wei Sun, May 18 2016

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4^k*m (k = 0,1,2,... and m = 7, 39, 87, 183, 231, 807, 879, 959, 1479, 2391, 2519, 2759, 4359, 10887).
See part (ii) of the conjecture in A269400 for similar conjectures.
For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

			a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2^2*1^2 + 3*1^2*1^2 + 2*1^2*1^2 = 3^2.
a(39) = 1 since 39 = 2^2 + 1^2 + 5^2 + 3^2 with 2^2*1^2 + 3*1^2*5^2 + 2*5^2*3^2 = 23^2.
a(87) = 1 since 87 = 2^2 + 1^2 + 1^2 + 9^2 with 2^2*1^2 + 3*1^2*1^2 + 2*1^2*9^2 = 13^2.
a(183) = 1 since 183 = 10^2 + 7^2 + 5^2 + 3^2 with 10^2*7^2 + 3*7^2*5^2 + 2*5^2*3^2 = 95^2.
a(231) = 1 since 231 = 10^2 + 1^2 + 9^2 + 7^2 with 10^2*1^2 + 3*1^2*9^2 + 2*9^2*7^2 = 91^2.
a(807) = 1 since 807 = 10^2 + 23^2 + 3^2 + 13^2 with 10^2*23^2 + 3*23^2*3^2 + 2*3^2*13^2 = 265^2.
a(879) = 1 since 879 = 14^2 + 11^2 + 21^2 + 11^2 with 14^2*11^2 + 3*11^2*21^2 + 2*21^2*11^2 = 539^2.
a(959) = 1 since 959 = 10^2 + 15^2 + 25^2 + 3^2 with 10^2*15^2 + 3*15^2*25^2 + 2*25^2*3^2 = 675^2.
a(1479) = 1 since 1479 = 34^2 + 11^2 + 11^2 + 9^2 with 34^2*11^2 + 3*11^2*11^2 + 2*11^2*9^2 = 451^2.
a(2391) = 1 since 2391 = 34^2 + 11^2 + 5^2 + 33^2 with 34^2*11^2 + 3*11^2*5^2 + 2*5^2*33^2 = 451^2.
a(2519) = 1 since 2519 = 42^2 + 1^2 + 27^2 + 5^2 with 42^2*1^2 + 3*1^2*27^2 + 2*27^2*5^2 = 201^2.
a(2759) = 1 since 2759 = 26^2 + 21^2 + 11^2 + 39^2 with 26^2*21^2 + 3*21^2*11^2 + 2*11^2*39^2 = 909^2.
a(4359) = 1 since 4359 = 46^2 + 19^2 + 19^2 + 39^2 with 46^2*19^2 + 3*19^2*19^2 + 2*19^2*39^2 = 1501^2.
a(10887) = 1 since 10887 = 31^2 + 85^2 + 51^2 + 10^2 with 31^2*85^2 + 3*85^2*51^2 + 2*51^2*10^2 = 7990^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x^2*y^2+3*y^2*z^2+2z^2*(n-x^2-y^2-z^2)],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,0,70}]

A273294 Least nonnegative integer m such that there are nonnegative integers x,y,z,w for which x^2 + y^2 + z^2 + w^2 = n and x + 3y + 5z = m^2.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 2, 3, 3, 3, 4, 0, 1, 2, 3, 2, 3, 3, 4, 4, 0, 1, 2, 3, 3, 4, 4, 2, 3, 3, 4, 0, 1, 2, 3, 4, 2, 3, 6, 4, 3, 3, 6, 4, 0, 1, 2, 2, 3, 5, 4, 4, 4, 3, 4, 5, 5, 3, 4, 0, 1, 2, 3, 4, 5, 4, 6, 4, 3, 4, 4, 4, 3, 4, 4, 2

Offset: 0

Zhi-Wei Sun, May 19 2016

nonn

Clearly, a(n) = 0 if n is a square. Part (i) of the conjecture in A271518 implies that a(n) always exists.

For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

			a(1) = 0 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 + 3*0 + 5*0 = 0^2.
a(2) = 1 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 + 3*0 + 5*0 = 1^2.
a(3) = 2 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1 + 3*1 + 5*0 = 2^2.
a(3812) = 11 since 3812 = 37^2 + 3^2 + 15^2 + 47^2 with 37 + 3*3 + 5*15 = 11^2.
a(3840) = 16 since 3840 = 48^2 + 16^2 + 32^2 + 16^2 with 48 + 3*16 + 5*32 = 16^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[m=0;Label[bb];Do[If[3y+5z<=m^2&&SQ[n-y^2-z^2-(m^2-3y-5z)^2],Print[n," ",m];Goto[aa]],{y,0,Sqrt[n]},{z,0,Sqrt[n-y^2]}];m=m+1;Goto[bb];Label[aa];Continue,{n,0,80}]

A273302 Least nonnegative integer x such that n = x^2 + y^2 + z^2 + w^2 for some nonnegative integer y,z,w with x + 3y + 5z a square.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 0, 0, 5, 4, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 4, 0, 0, 1, 1, 4, 0, 1, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 1, 0, 1, 1, 2, 3, 0, 0, 0, 1, 0, 0, 0, 3, 4

Offset: 0

Zhi-Wei Sun, May 19 2016

nonn

Clearly, a(n) = 0 if n is a square. Part (i) of the conjecture in A271518 implies that a(n) always exists.
Compare this sequence with A273294.
For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

			a(6) = 1 since 6 = 1^2 + 1^2 + 0^2 + 2^2 with 1 + 3*1 + 5*0 = 2^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 + 3*1 + 5*1 = 3^2.
a(15) = 2 since 15 = 2^2 + 3^2 + 1^2 + 1^2 with 2 + 3*3 + 5*1 = 4^2.
a(31) = 5 since 31 = 5^2 + 2^2 + 1^2 + 1^2 with 5 + 3*2 + 5*1 = 4^2.
a(32) = 4 since 32 = 4^2 + 0^2 + 0^2 + 4^2 with 4 + 3*0 + 5*0 = 2^2.
a(2384) = 24 since 2384 = 24^2 + 12^2 + 8^2 + 40^2 with 24 + 3*12 + 5*8 = 10^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+3y+5z],Print[n," ",x];Goto[aa]],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Label[aa];Continue,{n,0,80}]

cp's OEIS Frontend

A272888 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w*(x^2 + 8*y^2 - z^2) a square, where w,x,y are nonnegative integers and z is a positive integer.

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A272977 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 3*x^2*y + z^2*w a square, where w is a nonzero integer and x,y,z are nonnegative integers with x >= z.

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A273021 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x*y + y*z - z*w - w*x a square, where w is a positive integer and x,y,z are nonnegative integers with x <= y.

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A273108 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x + y)^2 + (4z)^2 a square, where x,y,z,w are nonnegative integers with x <= y > z.

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A273107 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (8*x+12*y)^2 + (15*z)^2 a square, where x,y,z,w are nonnegative integers with x+y > 0 and z > 0.

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A273110 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+4*y+4*z)^2 + (9*x+3*y+3*z)^2 a square, where x,y,z,w are nonnegative integers with y > 0 and y >= z <= w.

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A273134 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+8*y+8*z+15*w)^2+(6*(x+y+z+w))^2 a square, where x,y,z,w are nonnegative integers with y < z.

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A273278 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^2*y^2 + 3*y^2*z^2 + 2*z^2*w^2 is a square.

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A272888 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w(x^2 + 8y^2 - z^2) a square, where w,x,y are nonnegative integers and z is a positive integer.

A272977 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 3x^2y + z^2*w a square, where w is a nonzero integer and x,y,z are nonnegative integers with x >= z.

A273021 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 2xy + yz - zw - w*x a square, where w is a positive integer and x,y,z are nonnegative integers with x <= y.

A273107 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (8x+12y)^2 + (15*z)^2 a square, where x,y,z,w are nonnegative integers with x+y > 0 and z > 0.

A273110 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+4y+4z)^2 + (9x+3y+3*z)^2 a square, where x,y,z,w are nonnegative integers with y > 0 and y >= z <= w.

A273134 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+8y+8z+15w)^2+(6(x+y+z+w))^2 a square, where x,y,z,w are nonnegative integers with y < z.

A273278 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^2y^2 + 3y^2z^2 + 2z^2*w^2 is a square.

A273294 Least nonnegative integer m such that there are nonnegative integers x,y,z,w for which x^2 + y^2 + z^2 + w^2 = n and x + 3y + 5z = m^2.

A273302 Least nonnegative integer x such that n = x^2 + y^2 + z^2 + w^2 for some nonnegative integer y,z,w with x + 3y + 5z a square.