A270969 -id:A270969 - cp's OEIS frontend

A271518 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x + 3y + 5z a square, where w, x, y and z are nonnegative integers.

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

Offset: 0

Zhi-Wei Sun, Apr 09 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4^k*6 (k = 0,1,2,...), 16^k*m (k = 0,1,2,... and m = 5, 7, 8, 31, 43, 61, 116).
(ii) Any integer n > 15 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 6*x + 10*y + 12*z a square.
(iii) Each nonnegative integer n not among 7, 15, 23, 71, 97 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 2*x + 6*y + 10*z a square. Also, any nonnegative integer n not among 7, 43, 79 can be written as w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers and 3*x + 5*y + 6*z a square.
See also A271510 and A271513 for related conjectures.
a(n) > 0 verified for all n <= 3*10^7. - Zhi-Wei Sun, Nov 28 2016
Qing-Hu Hou at Tianjin Univ. has verified a(n) > 0 and parts (ii) and (iii) of the above conjecture for n up to 10^9. - Zhi-Wei Sun, Dec 04 2016
The conjecture that a(n) > 0 for all n = 0,1,2,... is called the 1-3-5-Conjecture and the author has announced a prize of 1350 US dollars for its solution. - Zhi-Wei Sun, Jan 17 2017
Qing-Hu Hou has finished his verification of a(n) > 0 for n up to 10^10. - Zhi-Wei Sun, Feb 17 2017
The 1-3-5 conjecture was finally proved by António Machiavelo and Nikolaos Tsopanidis in a JNT paper published in 2021. This is a great achivement! - Zhi-Wei Sun, Mar 31 2021

			a(5) = 1 since 5 = 2^2 + 1^2 + 0^2 + 0^2 with 1 + 3*0 + 5*0 = 1^2.
a(6) = 1 since 6 = 2^2 + 1^2 + 1^2 + 0^2 with 1 + 3*1 + 5*0 = 2^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 + 3*1 + 5*1 = 3^2.
a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.
a(24) = 1 since 24 = 4^2 + 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.
a(31) = 1 since 31 = 1^2 + 5^2 + 2^2 + 1^2 with 5 + 3*2 + 5*1 = 4^2.
a(43) = 1 since 43 = 1^2 + 1^2 + 5^2 + 4^2 with 1 + 3*5 + 5*4 = 6^2.
a(61) = 1 since 61 = 6^2 + 0^2 + 0^2 + 5^2 with 0 + 3*0 + 5*5 = 5^2.
a(116) = 1 since 116 = 10^2 + 4^2 + 0^2 + 0^2 with 4 + 3*0 + 5*0 = 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
António Machiavelo and Nikolaos Tsopanidis, Zhi-Wei Sun's 1-3-5 Conjecture and Variations, arXiv:2003.02592 [math.NT], 2020.
António Machiavelo and Nikolaos Tsopanidis, Zhi-Wei Sun's 1-3-5 Conjecture and Variations, J. Number Theory 222 (2021), 1-20.
António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, Report on Zhi-Wei Sun's "1-3-5 conjecture" and some of its refinements, arXiv:2005.13526 [math.NT], 2020.
António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, Report on Zhi-Wei Sun's 1-3-5 conjecture and some of its refinements, J. Number Theory 222 (2021), 21-29.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.NT], 2016.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. (See Conjecture 4.3(i) and Remark 4.3.)

Cf. A000118, A000290, A270969, A271510, A271513, A273294, A273302, A276533, A278560.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+3y+5z],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,0,80}]

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.

Original entry on oeis.org

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}]

A271510 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= 0, z >= 0 and w >= 0 such that x^2 + 8y^2 + 16z^2 is a square.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Apr 09 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 7, 23, 71, 77, 105, 191, 215, 311, 335, 2903, 4^k*q (k = 0,1,2,... and q = 6, 15, 47, 138).
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with x >= y >= 0, z >=0 and w >= 0 such that 4*x^2 + 21*y^2 + 24*z^2 (or 5*x^2 + 40*y^2 + 4*z^2, 20*x^2 + 85*y^2 +16*z^2, 25*x^2 + 480*y^2 + 96*z^2, 36*x^2 + 45*y^2 + 40*z^2, 40*x^2 + 72*y^2 + 9*z^2) is a square.
(iii) For any ordered pair (b, c) = (48, 112), (63, 7), (112, 1008), (136, 24), (136, 216), (360, 40), (840, 280), (1008, 112), each natural number can be written as x^2 + y^2 + z^2 + w^2 with x >= y >= 0, z >=0 and w >= 0 such that 9*x^2 + b*y^2 + c*z^2 is a square.
(iv) For any ordered pair (b, c) = (80, 25), (81, 48), (144, 9), (144, 153), (177, 48), each natural number can be written as x^2 + y^2 + z^2 + w^2 with x >= y >= 0, z >=0 and w >= 0 such that 16*x^2 + b*y^2 + c*z^2 is a square.
This conjecture is much stronger than Lagrange's four-square theorem. It is apparent that a(m^2*n) >= a(n) for all m,n = 1,2,3,....
See also A271513 and A271518 for related conjectures.
Conjectures (i), including the "a(n) = 1" part, (ii), (iii), and (iv) have been verified for n <= 10^9. - Mauro Fiorentini, Jun 19 2024

			a(6) = 1 since 6 = 1^2 + 1^2 + 0^2 + 2^2 with 1 = 1 and 1^2 + 8*1^2 + 16*0^2 = 3^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 = 1 and 1^2 + 8*1^2 + 16*1^2 = 5^2.
a(15) = 1 since 15 = 3^2 + 1^2 + 2^2 + 1^2 with 3 > 1 and 3^2 + 8*1^2 + 16*2^2 = 9^2.
a(23) = 1 since 23 = 3^2 + 1^2 + 2^2 + 3^2 with 3 > 1 and 3^2 + 8*1^2 + 16*2^2 = 9^2.
a(47) = 1 since 47 = 3^2 + 2^2 + 5^2 + 3^2 with 3 > 2 and 3^2 + 8*2^2 + 16*5^2 = 21^2.
a(71) = 1 since 71 = 7^2 + 2^2 + 3^2 + 3^2 with 7 > 2 and 7^2 + 8*2^2 + 16*3^2 = 15^2.
a(77) = 1 since 77 = 5^2 + 4^2 + 6^2 + 0^2 with 5 > 4 and 5^2 + 8*4^2 + 16*6^2 = 27^2.
a(105) = 1 since 105 = 6^2 + 2^2 + 4^2 + 7^2 with 6 > 2 and 6^2 + 8*2^2 + 16*4^2 = 18^2.
a(138) = 1 since 138 = 3^2 + 2^2 + 5^2 + 10^2 with 3 > 2 and 3^2 + 8*2^2 + 16*5^2 = 21^2.
a(191) = 1 since 191 = 9^2 + 3^2 + 1^2 + 10^2 with 9 > 3 and 9^2 + 8*3^2 + 16*1^2 = 13^2.
a(215) = 1 since 215 = 11^2 + 7^2 + 6^2 + 3^2 with 11 > 7 and 11^2 + 8*7^2 + 16*6^2 = 33^2.
a(311) = 1 since 311 = 15^2 + 6^2 + 1^2 + 7^2 with 15 > 6 and 15^2 + 8*6^2 + 16*1^2 = 23^2.
a(335) = 1 since 335 = 17^2 + 1^2 + 3^2 + 6^2 with 17 > 1 and 17^2 + 8*1^2 + 16*3^2 = 21^2.
a(2903) = 1 since 2903 = 49^2 + 14^2 + 15^2 + 9^2 with 49 > 14 and 49^2 + 8*14^2 + 16*15^2 = 87^2.

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. Also available from arXiv:1604.06723 [math.NT], 2016-2017.

Cf. A000118, A000290, A270969, A271513, A271518.

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

A271513 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 3x^2 + 4y^2 + 9*z^2 a square, where w, x, y and z are nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Apr 09 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 3, 11, 23, 43, 47, 67, 83, 107, 155, 323, 683, 803, 4^k*m (k = 0,1,2,... and m = 22, 38). [Conjecture verified for all natural numbers up to 10^9. - Mauro Fiorentini, Jul 04 2024]
(ii) Any natural number can be written as w^2 + x^2 + y^2 + z^2 with x, y, z integers and a*x^2 + b*y^2 + c*z^2 a square, whenever (a,b,c) is among the following triples: (1,3,12), (1,3,18), (1,3,21), (1,3,60), (1,5,15), (1,8,24), (1,12,15), (1,24,56), (1,24,72), (1,48,72), (1,48,168), (1,120,180), (1,192,288), (1,280,560), (3,9,13), (4,5,12), (4,5,60), (4,9,60), (4,12,21), (4,12,45), (4,12,69), (4,12,93), (4,12,237), (4,21,24), (4,21,36), (4,21,504), (4,24,93), (4,28,77), (4,45,120), (4,45,540), (4,45,600), (5,36,40), (7,9,126), (7,9,588), (8,16,73), (8,16,97), (8,49,112), (9,13,27), (9,16,24), (9,19,36), (9,21,91), (9,24,232), (9,28,63), (9,40,45), (9,40,56), (9,40,120), (9,45,115),(9,45,235), (12,13,24), (12,13,36), (12,36,37), (12,36,133), (13,36,72), (13,36,108), (15,24,25), (15,49,105), (16,17,48), (16,20,45), (16,21,84), (16,33,72), (16,33,176), (16,45,180), (16,48,57), (16,48,105), (16,48,233), (16,48,249), (19,45,57), (19,45,180), (21,25,35), (21,25,75), (21,28,36), (21,28,60), (21,43,105), (21,100,105),(24,25,72), (24,25,120), (24,48,97), (24,81,184), (24,120,145), (25,36,75), (25,40,56), (25,45,51), (25,45,99), (25,48,96), (25,48,144), (25,54,90), (25,75,81), (25,80,184), (25,96,120), (25,200,216), (28,33,36), (28,36,77), (28,72,189), (32,64,73), (33,36,220), (33,48,144), (33,72,256), (33,88,144), (36,45,100), (36,45,172), (37,81,243), (40,81,120), (40,81,240), (41,64,256), (45,48,76), (48,144,177), (49,56,64), (49,63,72), (55,141,165), (57,64,192), (60,105,196), (64,65,160), (72,73,144), (81,160,240), (85,140,196), (105,112,144), (112,144,153), (136,144,153), (144,145,240), (144,160,225),(148,189,252), (175,189,225). [Conjecture verified for all triples and all natural numbers up to 10^9. - Mauro Fiorentini, Jul 04 2024]
(iii) If a, b and c are positive integers such that any natural number can be written as w^2 + x^2 + y^2 + z^2 with x, y, z integers and a*x^2 + b*y^2 + c*z^2 a square, then a, b and c cannot be pairwise coprime.
This conjecture is stronger than Lagrange's four-square theorem. Moreover, there are many other suitable triples (a,b,c) for our purpose not listed in part (ii) of the conjecture. If a, b and c are positive integers such that any natural number can be written as w^2 + x^2 + y^2 + z^2 with x, y, z integers and a*x^2 + b*y^2 + c*z^2 a square, then one of a+b+c, 4*a+b+c, a+4*b+c and a+b+4*c must be a square since 2^2 + 1^2 + 1^2 + 1^2 is the unique way to express 7 as a sum of four squares.
Obviously, a(m^2*n) >= a(n) for all m,n = 1,2,3,....
See also A271510 and A271518 for related conjectures.

			a(3) = 1 since 3 = 0^2 + 1^2 + 1^2 + 1^2 with 3*1^2 + 4*1^2 + 9*1^2 = 4^2.
a(11) = 1 since 11 = 1^2 + 3^2 + 0^2 + 1^2 with 3*3^2 + 4*0^2 + 9*1^2 = 6^2.
a(22) = 1 since 22 = 4^2 + 2^2 + 1^2 + 1^2 with 3*2^2 + 4*1^2 + 9*1^2 = 5^2.
a(23) = 1 since 23 = 3^2 + 1^2 + 2^2 + 3^2 with 3*1^2 + 4*2^2 + 9*3^2 = 10^2.
a(38) = 1 since 38 = 0^2 + 6^2 + 1^2 + 1^2 with 3*6^2 + 4*1^2 + 9*1^2 = 11^2.
a(43) = 1 since 43 = 4^2 + 3^2 + 3^2 + 3^2 with 3*3^2 + 4*3^2 + 9*3^2 = 12^2.
a(47) = 1 since 47 = 3^2 + 6^2 + 1^2 + 1^2 with 3*6^2 + 4*1^2 + 9*1^2 = 11^2.
a(67) = 1 since 67 = 8^2 + 1^2 + 1^2 + 1^2 with 3*1^2 + 4*1^2 + 9*1^2 = 4^2.
a(83) = 1 since 83 = 0^2 + 9^2 + 1^2 + 1^2 with 3*9^2 + 4*1^2 + 9*1^2 = 16^2.
a(107) = 1 since 107 = 9^2 + 3^2 + 4^2 + 1^2 with 3*3^2 + 4*4^2 + 9*1^2 = 10^2.
a(155) = 1 since 155 = 0^2 + 9^2 + 5^2 + 7^2 with 3*9^2 + 4*5^2 + 9*7^2 = 28^2.
a(323) = 1 since 323 = 3^2 + 15^2 + 8^2 + 5^2 with 3*15^2 + 4*8^2 + 9*5^2 = 34^2.
a(683) = 1 since 683 = 15^2 + 11^2 + 16^2 + 9^2 with 3*11^2 + 4*16^2 + 9*9^2 = 46^2.
a(803) = 1 since 803 = 24^2 + 13^2 + 7^2 + 3^2 with 3*13^2 + 4*7^2 + 9*3^2 = 28^2.

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. Also available from arXiv:1604.06723 [math.NT], 2016-2017.

Cf. A000118, A000290, A270969, A271510, A271518.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[3x^2+4y^2+9z^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}]

A273429 Number of ordered ways to write n as x^6 + y^2 + z^2 + w^2, where x,y,z,w are nonnegative integers with y <= z <= w.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, May 22 2016

nonn

The author proved in arXiv:1604.06723 that for each c = 1, 4 any natural number can be written as c*x^6 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers. Thus a(n) > 0 for all n = 0,1,2,....
We note that a(n) = 1 for the following values of n not divisible by 2^6:  7, 8, 15, 16, 23, 24, 31, 32, 40, 47, 48, 56, 71, 79, 92, 112, 143, 176, 191, 240, 304, 368, 560, 624, 688, 752, 1072, 1136, 1456, 1520, 1840, 1904, 2608, 2672, 3760, 3824, 6512, 6896.
For more conjectural refinements of Lagrange's four-square theorem, one may consult the author's preprint arXiv:1604.06723.

			a(7) = 1 since 7 = 1^6 + 1^2 + 1^2 + 2^2 with 1 = 1 < 2.
a(8) = 1 since 8 = 0^6 + 0^2 + 2^2 + 2^2 with 0 < 2 = 2.
a(15) = 1 since 15 = 1^6 + 1^2 + 2^2 + 3^2 with 1 < 2 < 3.
a(16) = 1 since 16 = 0^6 + 0^2 + 0^2 + 4^2 with 0 = 0 < 4.
a(56) = 1 since 56 = 0^6 + 2^2 + 4^2 + 6^2 with 2 < 4 < 6.
a(71) = 1 since 71 = 1^6 + 3^2 + 5^2 + 6^2 with 3 < 5 < 6.
a(79) = 1 since 79 = 1^6 + 2^2 + 5^2 + 7^2 with 2 < 5 < 7.
a(92) = 1 since 92 = 1^6 + 1^2 + 3^2 + 9^2 with 1 < 3 < 9.
a(143) = 1 since 143 = 1^6 + 5^2 + 6^2 + 9^2 with 5 < 6 < 9.
a(191) = 1 since 191 = 1^6 + 3^2 + 9^2 + 10^2 with 3 < 9 < 10.
a(624) = 1 since 624 = 2^6 + 4^2 + 12^2 + 20^2 with 4 < 12 < 20.
a(2672) = 1 since 2672 = 2^6 + 4^2 + 36^2 + 36^2 with 4 < 36 = 36.
a(3760) = 1 since 3760 = 0^6 + 4^2 + 12^2 + 60^2 with 4 < 12 < 60.
a(3824) = 1 since 3824 = 2^6 + 4^2 + 12^2 + 60^2 with 4 < 12 < 60.
a(6512) = 1 since 6512 = 2^6 + 12^2 + 52^2 + 60^2 with 12 < 52 < 60.
a(6896) = 1 since 6896 = 2^6 + 36^2 + 44^2 + 60^2 with 36 < 44 < 60.

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.NT], 2016-2017.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Cf. A000118, A000290, A001014, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273432, A273568.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^6-y^2-z^2],r=r+1],{x,0,n^(1/6)},{y,0,Sqrt[(n-x^6)/3]},{z,y,Sqrt[(n-x^6-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}]

A273432 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x + y - z a nonnegative cube, where x,y,z,w are nonnegative integers with y <= z.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, May 22 2016

nonn

Conjecture: (i) For each c = 1, 2, 4 and n = 0,1,2,..., we can write n as x^2 + y^2 + z^2 + w^2 with c*(2x+y-z) a nonnegative cube, where x,y,z,w are nonnegative integers with y <= z.
(ii) Each n = 0,1,2,.... can be written as x^2 + y^2 + z^2 + w^2 with x-y+z a nonnegative cube, where x,y,z,w are integers with x >= y >= 0 and x >= |z|.
The author proved in arXiv:1604.06723 that for each a = 1, 2 any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + y + a*z is a cube.
See also A273458 for a similar conjecture.
For more conjectural refinements of Lagrange's four-square theorem, see the author's preprint arXiv:1604.06723.

			a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0 and 2*0 + 0 - 0 = 0^3.
a(4) = 1 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 = 0 and 2*0 + 0 - 0 = 0^3.
a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 2 = 2 and 2*0 + 2 - 2 = 0^3.
a(10) = 1 since 10 = 1^2 + 1^2 + 2^2 + 2^2 with 1 < 2 and 2*1 + 1 - 2 = 1^3.
a(13) = 1 since 13 = 2^2 + 0^2 + 3^2 + 0^2 with 0 < 3 and 2*2 + 0 - 3 = 1^3.
a(23) = 1 since 23 = 1^2 + 2^2 + 3^2 + 3^2 with 2 < 3 and 2*1 + 2 - 3 = 1^3.
a(26) = 1 since 26 = 1^2 + 3^2 + 4^2 + 0^2 with 3 < 4 and 2*1 + 3 - 4 = 1^3.
a(28) = 1 since 28 = 4^2 + 2^2 + 2^2 + 2^2 with 2 = 2 and 2*4 + 2 - 2 = 2^3.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 2 = 2 and 2*4 + 2 - 2 = 2^3.
a(104) = 1 since 104 = 4^2 + 6^2 + 6^2 + 4^2 with 6 = 6 and 2*4 + 6 - 6 = 2^3.
a(138) = 1 since 138 = 3^2 + 5^2 + 10^2 + 2^2 with 5 < 10 and 2*3 + 5 - 10 =1^3.
a(200) = 1 since 200 = 0^2 + 10^2 + 10^2 + 0^2 with 10 = 10 and 2*0 + 10 - 10 = 0^3.
a(296) = 1 since 296 = 8^2 + 6^2 + 14^2 + 0^2 with 6 < 14 and 2*8 + 6 - 14 = 2^3.
a(328) = 1 since 328 = 0^2 + 6^2 + 6^2 + 16^2 with 6 = 6 and 2*0 + 6 - 6 = 0^3.
a(520) = 1 since 520 = 4^2 + 2^2 + 10^2 + 20^2 with 2 < 10 and 2*4 + 2 - 10 = 0^3.
a(776) = 1 since 776 = 0^2 + 10^2 + 10^2 + 24^2 with 10 = 10 and 2*0 + 10 - 10 = 0^3.
a(1832) = 1 since 1832 = 4^2 + 30^2 + 30^2 + 4^2 with 30 = 30 and 2*4 + 30 - 30 = 2^3.
a(2976) = 1 since 2976 = 20^2 + 16^2 + 48^2 + 4^2 with 16 < 48 and 2*20 + 16 - 48 = 2^3.

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, A000578, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273429, A273458, A273568.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&CQ[2x+y-z],r=r+1],{x,0,n^(1/2)},{y,0,Sqrt[(n-x^2)/2]},{z,y,Min[2x+y,Sqrt[n-x^2-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}]

cp's OEIS Frontend

A271518 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x + 3*y + 5*z a square, where w, x, y and z are nonnegative integers.

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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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A271510 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= 0, z >= 0 and w >= 0 such that x^2 + 8*y^2 + 16*z^2 is a square.

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A271513 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 3*x^2 + 4*y^2 + 9*z^2 a square, where w, x, y and z are nonnegative integers.

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A273429 Number of ordered ways to write n as x^6 + y^2 + z^2 + w^2, where x,y,z,w are nonnegative integers with y <= z <= w.

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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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A273432 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x + y - z a nonnegative cube, where x,y,z,w are nonnegative integers with y <= z.

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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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A271518 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x + 3y + 5z a square, where w, x, y and z are nonnegative integers.

A271510 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= 0, z >= 0 and w >= 0 such that x^2 + 8y^2 + 16z^2 is a square.

A271513 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 3x^2 + 4y^2 + 9*z^2 a square, where w, x, y and z are nonnegative integers.

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.