A281939 -id:A281939 - cp's OEIS frontend

A281976 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and x + 24*y are squares.

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

Offset: 0

Zhi-Wei Sun, Feb 04 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 8, 12, 23, 24, 47, 71, 168, 344, 632, 1724).
By the linked JNT paper, any nonnegative integer can be written as the sum of a fourth power and three squares.
We have verified a(n) > 0 for all n = 0..10^7.
See also A281977, A282013 and A282014 for similar conjectures.
a(n) <= A273404(n). Starts to differ from A273404 at n=145. - R. J. Mathar, Feb 12 2017
Qing-Hu Hou at Tianjin Univ. has verified a(n) > 0 for all n = 0..10^10.
I would like to offer 2400 US dollars for the first proof of my conjecture that a(n) > 0 for any nonnegative integer n. - Zhi-Wei Sun, Feb 14 2017

			a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and 0 + 24*0 = 0^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 = 1^2 and 1 + 24*1 = 5^2.
a(23) = 1 since 23 = 1^2 + 2^2 + 3^2 + 3^2 with 1 = 1^2 and 1 + 24*2 = 7^2.
a(24) = 1 since 24 = 4^2 + 0^2 + 2^2 + 2^2 with 4 = 2^2 and 4 + 24*0 = 2^2.
a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 = 1^2 and 1 + 24*1 = 5^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 = 1^2 and 1 + 24*5 = 11^2.
a(168) = 1 since 168 = 4^2 + 4^2 + 6^2 + 10^2 with 4 = 2^2 and 4 + 24*4 = 10^2.
a(344) = 1 since 344 = 4^2 + 0^2 + 2^2 + 18^2 with 4 = 2^2 and 4 + 24*0 = 2^2.
a(632) = 1 since 632 = 0^2 + 6^2 + 14^2 + 20^2 with 0 = 0^2 and 0 + 24*6 = 12^2.
a(1724) = 1 since 1724 = 25^2 + 1^2 + 3^2 + 33^2 with 25 = 5^2 and 25 + 24*1 = 7^2.

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.
Zhi-Wei Sun, The 24-conjecture with $2400 prize, a message to Number Theory List, Feb. 14, 2017.

Cf. A000118, A000290, A000583, A270969, A273404, A281939, A281941, A281975, A281977, A281980, A282013, A282014.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[x^2+24y],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[(n-x^4-y^2)/2]}];Print[n," ",r];Continue,{n,0,80}]

A281977 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and -7x - 8y + 8z + 16w are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,....
The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.
We have verified the conjecture for all n = 0..10^6.
See also A281976, A282013 and A282014 for similar conjectures.
Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^8. - Zhi-Wei Sun, Jun 02 2019

			a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0^2 and -7*0 - 8*0 + 8*0 + 16*1 = 4^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 3^2 + 1^2 with 1 = 1^2 and -7*1 - 8*1 + 8*3 + 16*1 = 5^2.
a(17) = 1 since 17 = 1^2 + 0^2 + 4^2 + 0^2 with 1 = 1^2 and -7*1 - 8*0 + 8*4 + 16*0 = 5^2.
a(28) = 1 since 28 = 4^2 + 2^2 + 2^2 + 2^2 with 4 = 2^2 and -7*4 - 8*2 + 8*2 + 16*2 = 2^2.
a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 1 = 1^2 and -7*1 - 8*1 + 8*2 + 16*5 = 9^2.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 4 = 2^2 and -7*4 -8*2 + 8*2 + 16*4 = 6^2.
a(41) = 1 since 41 = 1^2 + 2^2 + 6^2 + 0^2 with 1 = 1^2 and -7*1 - 8*2 + 8*6 + 16*0 = 5^2.
a(49) = 1 since 49 = 0^2 + 6^2 + 2^2 + 3^2 with 0 = 0^2 and -7*0 - 8*6 + 8*2 + 16*3 = 4^2.
a(241) = 1 since 241 = 9^2 + 4^2 + 12^2 + 0^2 with 9 = 3^2 and -7*9 - 8*4 + 8*12 + 16*0 = 1^2.
a(433) = 1 since 433 = 16^2 + 8^2 + 8^2 + 7^2 with 16 = 4^2 and -7*16 - 8*8 + 8*8 + 16*7 = 0^2.
a(1113) = 1 since 1113 = 1^2 + 30^2 + 4^2 + 14^2 with 1 = 1^2 and -7*1 - 8*30 + 8*4 + 16*14 = 3^2.
a(1521) = 1 since 1521 = 0^2 + 22^2 + 14^2 + 29^2 with 0 = 0^2 and -7*0 - 8*22 + 8*14 + 16*29 = 20^2.

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.

Cf. A000118, A000290, A270969, A281939, A281941, A281975, A281976, A282013, A282014.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[16*Sqrt[n-x^4-y^2-z^2]+8z-8y-7x^2],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A281941 Number of ways to write n as w^2 + x^2 + y^2 + z^2 with w and w + x + y + z both squares, where w,x,y,z are integers with |x| <= |y| <= |z|.

Original entry on oeis.org

1, 2, 3, 5, 4, 1, 5, 2, 3, 4, 5, 1, 3, 1, 4, 2, 2, 1, 10, 5, 2, 2, 6, 1, 7, 7, 5, 7, 4, 3, 7, 1, 3, 12, 9, 4, 2, 2, 5, 3, 5, 5, 9, 10, 1, 5, 5, 1, 5, 3, 6, 8, 2, 4, 9, 4, 4, 8, 5, 3, 3, 4, 5, 3, 4, 5, 10, 4, 1, 5, 7, 1, 7, 10, 6, 8, 3, 2, 10, 2, 1

Offset: 0

Zhi-Wei Sun, Feb 02 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
(ii) Any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x - y + z and z + w both squares, where x,w are integers and y,z are nonnegative integers.
The author has proved that every n = 0,1,2,... is the sum of a fourth power and three squares. Y.-C. Sun and the author have shown that any nonnegative integer can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that w + x + y + z is a square.

			a(5) = 1 since 5 = 0^2 + 0^2 + (-1)^2 + 2^2 with 0 = 0^2 and 0 + 0 + (-1) + 2 = 1^2.
a(23) = 1 since 23 = 1^2 + 2^2 + 3^2 + 3^2 with 1 = 1^2 and 1 + 2 + 3 + 3 = 3^2.
a(47) = 1 since 47 = 1^2 + (-1)^2 + 3^2 + 6^2 with 1 = 1^2 and 1 + (-1) + 3 + 6 = 3^2.
a(157) = 1 since 157 = 4^2 + (-2)^2 + (-4)^2 + 11^2 with 4 = 2^2 and 4 + (-2) + (-4) + 11 = 3^2.
a(284) = 1 since 284 = 9^2 + 3^2 + 5^2 + (-13)^2 with 9 = 3^2 and 9 + 3 + 5 + (-13) = 2^2.
a(628) = 1 since 628 = 9^2 + (-5)^2 + (-9)^2 + 21^2 with 9 = 3^2 and 9 + (-5) + (-9) + 21 = 4^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Yu-Chen Sun and Zhi-Wei Sun, Some refinements of Lagrange's four-square theorem, arXiv:1605.03074 [math.NT], 2016-2017.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Cf. A000118, A000290, A270969, A272620, A281939.

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

A282013 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 49x + 48(y-z) are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 5, 8, 14, 31, 47, 71, 79, 143, 248, 463, 1039).
The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.
We have verified a(n) > 0 for all n = 0..10^7.
See also A281976, A281977 and A282014 for similar conjectures.
Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Jun 02 2019

			a(5) = 1 since 5 = 1^2 + 0^2 + 0^2 + 2^2 with 1 = 1^2 and 49*1 + 48*(0-0) = 7^2.
a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 0 = 0^2 and 49*0 + 48*(2-2) = 0^2.
a(14) = 1 since 14 = 1^2 + 2^2 + 3^2 + 0^2 with 1 = 1^2 and 49*1 + 48*(2-3) = 1^2.
a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 1 = 1^2 and 49*1 + 48*(1-2) = 1^2.
a(47) = 1 since 47 = 1^2 + 6^2 + 1^2 + 3^2 with 1 = 1^2 and 49*1 + 48*(6-1) = 17^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 6^2 + 3^2 with 1 = 1^2 and 49*1 + 48*(5-6) = 1^2.
a(79) = 1 since 79 = 1^2 + 7^2 + 2^2 + 5^2 with 1 = 1^2 and 49*1 + 48*(7-2) = 17^2.
a(143) = 1 since 143 = 1^2 + 5^2 + 6^2 + 9^2 with 1 = 1^2 and 49*1 + 48*(5-6) = 1^2.
a(248) = 1 since 248 = 4^2 + 6^2 + 0^2 + 14^2 with 4 = 2^2 and 49*4 + 48*(6-0) = 22^2.
a(463) = 1 since 463 = 9^2 + 6^2 + 15^2 + 11^2 with 9 = 3^2 and 49*9 + 48*(6-15) = 3^2.
a(1039) = 1 since 1039 = 1^2 + 22^2 + 23^2 + 5^2 with 1 = 1^2 and 49*1 + 48*(22-23) = 1^2.

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.

Cf. A000118, A000290, A270969, A271518, A281939, A281941, A281975, A281976, A281977, A282014.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[49x^2+48(y-z)],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A282014 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 121x + 48(y-z) are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 8, 29, 31, 40, 94, 104, 143, 319, 671).
The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.
We have verified a(n) > 0 for all n = 0..10^7.
See also A281976, A281977 and A282013 for similar conjectures.
Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Jun 02 2019

			a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(2-2) = 0^2.
a(29) = 1 since 29 = 0^2 + 5^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(5-2) = 12^2.
a(31) = 1 since 31 = 1^2 + 2^2 + 1^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(2-1) = 13^2.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(2-2) = 22^2.
a(94) = 1 since 94 = 0^2 + 6^2 + 3^2 + 7^2 with 0 = 0^2 and 121*0 + 48*(6-3) = 12^2.
a(104) = 1 since 104 = 4^2 + 6^2 + 6^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(6-6) = 22^2.
a(143) = 1 since 143 = 1^2 + 6^2 + 5^2 + 9^2 with 1 = 1^2 and 121*1 + 48*(6-5) = 13^2.
a(319) = 1 since 319 = 1^2 + 17^2 + 2^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(17-2) = 29^2.
a(671) = 1 since 671 = 9^2 + 5^2 + 23^2 + 6^2 with 9 = 3^2 and 121*9 + 48*(5-23) = 15^2.

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.

Cf. A000118, A000290, A270969, A271518, A281939, A281941, A281975, A281976, A281977, A282013.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[121x^2+48(y-z)],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A283170 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 2y and z + 2w both squares, where x and w are nonnegative integers, and y and z are integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Mar 02 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
(ii) Any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x + 3*y and z + 3*w both squares, where x,y,z are integers and w is a nonnegative integer.
(iii) Every nonnegative integer can be expressed as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z + 3*w are squares.
(vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is a square and |2*z-w| is twice a square. Also, each nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is twice a square and |2*z-w| is a square.
(v) Every 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 |(x-2*y)*(z-2*w)| is twice a square. Also, any positive integer n can be written as x^2 + y^2 + z^2 + w^2 with x a positive integer and y,z,w nonnegative integers such that (2*x+y)*(2*z-w) is twice a square.
(vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z^2 - w^2 (or z^2 + 8*w^2, or 7*z^2 + 9*w^2) are squares.
(vii) Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 64*z^2 - 84*z*w + 21*w^2 (or 81*z^2 - 112*z*w + 56*w^2) are squares.
By the linked JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + 2*y is a square.

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.

Cf. A000118, A000290, A271518, A281976, A281939, A282561, A282562.

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

a(3) = 1 since 3 = 1^2 + 0^2 + (-1)^2 + 1^2 with 1 + 2*0 = 1^2 and (-1)+2*1 = 1^2.
a(4) = 2 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 + 2*0 = 0^2 and 0 + 2*2 = 2^2, and also 4 = 0^2 + 2^2 + 0^2 + 0^2 with 0 + 2*2 = 2^2 and 0 + 2*0 = 0^2.
a(8) = 1 since 8 = 0^2 + 2^2 + 0^2 + 2^2 with 0 + 2*2 = 2^2 and 0 + 2*2 = 2^2.
a(11) = 1 since 11 = 3^2 + (-1)^2 + 1^2 + 0^2 with 3 + 2*(-1) = 1^2 and 1 + 2*0 = 1^2.
a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^2 with 3 + 2*(-1) = 1^2 and (-1) + 2*1 = 1^2.
a(28) = 1 since 28 = 3^2 + (-1)^2 + 3^2 + 3^2 with 3 + 2*(-1) = 1^2 and 3 + 2*3 = 3^2.
a(40) = 1 since 40 = 4^2 + (-2)^2 + (-4)^2 + 2^2 with 4 + 2*(-2) = 0^2 and (-4) + 2*2 = 0^2.
a(41) = 1 since 41 = 6^2 + (-1)^2 + 0^2 + 2^2 with 6 + 2*(-1) = 2^2 and 0 + 2*2 = 2^2.
a(332) = 1 since 332 = 11^2 + 7^2 + (-9)^2 + 9^2 with 11 + 2*7 = 5^2 and (-9) + 2*9 = 3^2.
a(443) = 1 since 443 = 19^2 + (-9)^2 + 1^2 + 0^2 with 19 + 2*(-9) = 1^2 and 1 + 2*0 = 1^2.
a(488) = 1 since 488 = 12^2 + 2^2 + (-12)^2 + 14^2 with 12 + 2*2 = 4^2 and (-12) + 2*14 = 4^2.
a(808) = 1 since 808 = 8^2 + 14^2 + (-8)^2 + 22^2 with 8 + 2*14 = 6^2 and (-8) + 2*22 = 6^2.
a(892) = 1 since 892 = 27^2 + (-1)^2 + (-9)^2 + 9^2 with 27 + 2*(-1) = 5^2 and (-9) + 2*9 = 3^2.

A281975 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and |x-y| are squares.

Original entry on oeis.org

1, 3, 4, 3, 2, 4, 4, 2, 1, 4, 5, 3, 1, 2, 3, 2, 3, 5, 7, 3, 4, 5, 2, 1, 2, 6, 9, 6, 2, 4, 6, 3, 4, 6, 8, 4, 6, 5, 4, 3, 2, 11, 10, 4, 1, 7, 5, 1, 3, 3, 11, 9, 7, 5, 6, 4, 2, 7, 5, 4, 2, 6, 4, 3, 2, 7, 15, 4, 4, 6, 5, 1, 2, 6, 7, 7, 3, 6, 4, 2, 4

Offset: 0

Zhi-Wei Sun, Feb 03 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
(ii) Each nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that |x-y| and 2*(y-z) (or 2*(z-y)) are both squares.
(iii) For each ordered pair (a,b) = (2,1), (3,1), (9,5), (14,10), any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x and |a*x-b*y| are both squares.
The author has proved that 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 x (or x-y, or 2(x-y)) is a square.
See also A281976 and A281977 for similar conjectures.

			a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and |0-0| = 0^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 = 1^2 and |1-1| = 0^2.
a(44) = 1 since 44 = 1^2 + 5^2 + 3^2 + 3^2 with 1 = 1^2 and |1-5| = 2^2.
a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 = 1^2 and |1-1| = 0^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 = 1^2 and |1-5| = 2^2.
a(95) = 1 since 95 = 1^2 + 2^2 + 3^2 + 9^2 with 1 = 1^2 and |1-2| = 1^2.
a(140) = 1 since 140 = 9^2 + 5^2 + 3^2 + 5^2 with 9 = 3^2 and |9-5| = 2^2.
a(428) = 1 since 428 = 9^2 + 13^2 + 3^2 + 13^2 with 9 = 3^2 and |9-13| = 2^2.
a(568) = 1 since 568 = 4^2 + 8^2 + 2^2 + 22^2 with 4 = 2^2 and |4-8| = 2^2.
a(632) = 1 since 632 = 16^2 + 12^2 + 6^2 + 14^2 with 16 = 4^2 and |16-12| = 2^2.
a(1144) = 1 since 1144 = 16^2 + 20^2 + 2^2 + 22^2 with 16 = 4^2 and |16-20| = 2^2.
a(1544) = 1 since 1544 = 0^2 + 0^2 + 10^2 + 38^2 with 0 = 0^2 and |0-0| = 0^2.

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.

Cf. A000118, A000290, A270969, A271775, A281939, A281941, A281976, A281977.

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

A283196 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 2x + y and 2x + z both squares, where x,y,z are integers with |y| <= |z|, and w is a positive integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 1, 1, 8, 1, 1, 6, 1, 3, 1, 3, 9, 2, 3, 3, 4, 4, 1, 7, 5, 2, 4, 3, 3, 6, 1, 5, 7, 1, 5, 4, 6, 4, 3, 2, 8, 3, 2, 11, 2, 6, 1, 6, 5, 1, 9, 4, 7, 11, 1, 3, 16, 1, 2, 5, 3, 14, 2, 7, 7, 4, 6, 3, 12, 6, 3, 8, 5, 2, 3, 5, 5, 9, 2

Offset: 1

Zhi-Wei Sun, Mar 02 2017

nonn

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Any positive integer n can be written as x^2 + y^2 + z^2 + w^2 such that both x + y and x + 2*z are squares, where x,y,z,w are integers with x >= 0 and w > 0.
(iii)  Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with 2*x + 2*y and 2*x + z both squares, where x,y,z,w are integers with x*y <= 0.
(iv) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with 2*|x-y| and 2*x + z both squares, where x,y,z,w are integers with x >= 0 and y >= 0.
By the linked JNT paper, any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that 2*x + y is a square, and also we can write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x - y (or 2*x - 2*y) is a square.

			a(2) = 1 since 2 = 0^2 + 0^2 + 1^2 + 1^2 with 2*0 + 0 = 0^2 and 2*0 + 1 = 1^2.
a(14) = 1 since 14 = 2^2 + 0^2 + (-3)^2 + 1^2 with 2*2 + 0 = 2^2 and 2*2 + (-3) = 1^2.
a(59) = 1 since 59 = 3^2 + 3^2 + (-5)^2 + 4^2 with 2*3 + 3 = 3^2 and 2*3 + (-5) = 1^2.
a(88) = 1 since 88 = (-2)^2 + 4^2 + 8^2 + 2^2 with 2*(-2) + 4 = 0^2 and 2*(-2) + 8 = 2^2.
a(131) = 1 since 131 = 0^2 + 1^2 + 9^2 + 7^2 with 2*0 + 1 = 1^2 and 2*0 + 9 = 3^2.
a(219) = 1 since 219 = 8^2 + (-7)^2 + 9^2 + 5^2 with 2*8 + (-7) = 3^2 and 2*8 + 9 = 5^2.
a(249) = 1 since 249 = (-4)^2 + 8^2 + 12^2 + 5^2 with 2*(-4) + 8 = 0^2 and 2*(-4) + 12 = 2^2.
a(312) = 1 since 312 = 6^2 + 4^2 + (-8)^2 + 14^2 with 2*6 + 4 = 4^2 and 2*6 + (-8) = 2^2.
a(323) = 1 since 323 = 9^2 + 7^2 + 7^2 + 12^2 with 2*9 + 7 = 5^2.
a(536) = 1 since 536 = (-6)^2 + 12^2 + 16^2 + 10^2 with 2*(-6) + 12 = 0^2 and 2*(-6) + 16 = 2^2.
a(888) = 1 since 888 = 14^2 + 8^2 + (-12)^2 + 22^2 with 2*14 + 8 = 6^2 and 2*14 + (-12) = 4^2.
a(1464) = 1 since 1464 = 2^2 + 0^2 + (-4)^2 + 38^2 with 2*2 + 0 = 2^2 and 2*2 + (-4) = 0^2.
a(4152) = 1 since 4152 = 30^2 + 4^2 + (-56)^2 + 10^2 with 2*30 + 4 = 8^2 and 2*30 + (-56) = 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..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.

Cf. A000118, A000290, A271518, A281975, A281976, A281939, A283170.

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

A283205 Numbers of the form x^2 + y^2 with x and y integers such that x + 3*y is a square.

Original entry on oeis.org

0, 1, 2, 5, 8, 9, 10, 13, 16, 17, 25, 26, 29, 32, 34, 37, 40, 50, 53, 58, 61, 65, 73, 74, 80, 81, 85, 90, 104, 109, 117, 125, 128, 130, 136, 137, 144, 145, 146, 160, 162, 170, 178, 185, 193, 202, 208, 221, 229, 232, 241, 245, 250, 256, 257, 265, 269, 272, 274, 281

Offset: 1

Zhi-Wei Sun, Mar 03 2017

nonn

This sequence is interesting since part (ii) of the conjecture in A283170 implies that each n = 0,1,2,... can be expressed as the sum of two terms of the current sequence.

Clearly, the sequence is a subsequence of A001481.

			a(1) = 0 since 0 = 0^2 + 0^2 with 0 + 3*0 = 0^2.
a(2) = 1 since 1 = 1^2 + 0^2 with 1 + 3*0 = 1^2.
a(3) = 2 since 2 = 1^2 + 1^2 with 1 + 3*1 = 2^2.
a(4) = 5 since 5 = (-2)^2 + 1^2 with (-2) + 3*1 = 1^2.
a(5) = 8 since 8 = (-2)^2 + 2^2 with (-2) + 3*2 = 2^2.
a(6) = 9 since 9 = 0^2 + 3^2 with 0 + 3*3 = 3^2.
a(7) = 10 since 10 = 3^2 + (-1)^2 with 3 + 3*(-1) = 0^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..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.

Cf. A000290, A001481, A281939, A283170, A283204.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
n=0;Do[Do[If[SQ[m-x^2],Do[If[SQ[(-1)^i*x+3(-1)^j*Sqrt[m-x^2]],n=n+1;Print[n," ",m];Goto[aa]],{i,0,Min[x,1]},{j,0,Min[Sqrt[m-x^2],1]}]],{x,0,Sqrt[m]}];Label[aa];Continue,{m,0,281}]

A282562 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 9z^2 + 666zw + w^2 are squares.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 5, 2, 3, 5, 2, 2, 1, 5, 6, 2, 2, 5, 5, 1, 5, 7, 8, 1, 3, 4, 2, 4, 3, 9, 7, 5, 2, 5, 5, 4, 8, 3, 9, 3, 4, 4, 5, 3, 1, 7, 6, 3, 2, 13, 7, 5, 5, 5, 8, 3, 2, 3, 5, 2, 1, 6, 10, 6, 4, 8, 9, 1, 7, 8, 11, 3, 3, 6, 6, 5, 3, 11, 4, 4, 4

Offset: 0

Zhi-Wei Sun, Feb 18 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 12, 19, 23, 44, 60, 67, 139, 140, 248, 264, 347, 427, 499, 636, 1388, 1867, 1964, 4843).
By the linked JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z*(z-w) = 0. Whether z = 0 or z = w, the number 9*z^2 + 666*z*w + w^2 is definitely a square.
See also A282561 for a similar conjecture.

			a(12) = 1 since 12 = 2^2 + 0^2 + 2^2 + 2^2 with 2*2 - 0 = 2^2 and 9*2^2 + 666*2*2 + 2^2 = 52^2.
a(19) = 1 since 19 = 1^2 + 1^2 + 4^2 + 1^2 with 2*1 - 1 = 1^2 and 9*4^2 + 666*4*1 + 1^2 = 53^2.
a(44) = 1 since 44 = 5^2 + 1^2 + 3^2 + 3^2 with 2*5 - 1 = 3^2 and 9*3^2 + 666*3*3 + 3^2 = 78^2.
a(60) = 1 since 60 = 3^2 + 5^2 + 1^2 + 5^2 with 2*3 - 5 = 1^2 and 9*1^2 + 666*1*5 + 5^2 = 58^2.
a(67) = 1 since 67 = 4^2 + 7^2 + 1^2 + 1^2 with 2*4 - 7 = 1^2 and 9*1^2 + 666*1*1 + 1^2 = 26^2.
a(139) = 1 since 139 = 8^2 + 7^2 + 1^2 + 5^2 with 2*8 - 7 = 3^2 and 9*1^2 + 666*1*5 + 5^2 = 58^2.
a(140) = 1 since 140 = 3^2 + 5^2 + 5^2 + 9^2 with 2*3 - 5 = 1^2 and 9*5^2 + 666*5*9 + 9^2 = 174^2.
a(264) = 1 since 264 = 8^2 + 0^2 + 10^2 + 10^2 with 2*8 - 0 = 4^2 and 9*10^2 + 666*10*10 + 10^2 = 260^2.
a(499) = 1 since 499 = 7^2 + 5^2 + 20^2 + 5^2 with 2*7 - 5 = 3^2 and 9*20^2 + 666*20*5 + 5^2 = 265^2.
a(1388) = 1 since 1388 = 15^2 + 21^2 + 19^2 + 19^2 with 2*15 - 21 = 3^2 and 9*19^2 + 666*19*19 + 19^2 = 494^2.
a(1867) = 1 since 1867 = 16^2 + 31^2 + 5^2 + 25^2 with 2*16 - 31 = 1^2 and 9*5^2 + 666*5*25 + 25^2 = 290^2.
a(4843) = 1 since 4843 = 11^2 + 13^2 + 52^2 + 43^2 with 2*11 - 13 = 3^2 and 9*52^2 + 666*52*43 + 43^2 = 1231^2.

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.

Cf. A000118, A000290, A271518, A281939, A281976, A282463, A282494, A282495, A282545, A282561.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[2x-y],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[9z^2+666z*Sqrt[n-x^2-y^2-z^2]+(n-x^2-y^2-z^2)],r=r+1],{z,0,Sqrt[n-x^2-y^2]}]],{y,0,Sqrt[4n/5]},{x,Ceiling[y/2],Sqrt[n-y^2]}];Print[n," ",r];Continue,{n,0,80}]

cp's OEIS Frontend

A281976 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and x + 24*y are squares.

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A281977 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and -7*x - 8*y + 8*z + 16*w are squares.

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A281941 Number of ways to write n as w^2 + x^2 + y^2 + z^2 with w and w + x + y + z both squares, where w,x,y,z are integers with |x| <= |y| <= |z|.

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A282013 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 49*x + 48*(y-z) are squares.

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A282014 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 121*x + 48*(y-z) are squares.

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A283170 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 2*y and z + 2*w both squares, where x and w are nonnegative integers, and y and z are integers.

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A281975 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and |x-y| are squares.

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A283196 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x + y and 2*x + z both squares, where x,y,z are integers with |y| <= |z|, and w is a positive integer.

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A281977 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and -7x - 8y + 8z + 16w are squares.

A282013 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 49x + 48(y-z) are squares.

A282014 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 121x + 48(y-z) are squares.

A283170 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 2y and z + 2w both squares, where x and w are nonnegative integers, and y and z are integers.

A283196 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 2x + y and 2x + z both squares, where x,y,z are integers with |y| <= |z|, and w is a positive integer.

A282562 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 9z^2 + 666zw + w^2 are squares.