A271513 -id:A271513 - 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}]

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}]

A271608 Number of ordered ways to write n as pen(u) + pen(v) + pen(x) + pen(y) + pen(z) with u,v,x,y,z nonnegative integers such that u + 2v + 4x + 5y + 6z is a pentagonal number, where pen(k) denotes the pentagonal number k*(3k-1)/2.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 3, 3, 3, 3, 3, 3, 4, 2, 6, 4, 2, 1, 1, 8, 4, 5, 2, 2, 7, 10, 9, 2, 3, 4, 5, 6, 6, 5, 2, 7, 11, 11, 4, 1, 5, 8, 13, 8, 6, 5, 3, 8, 8, 12, 7, 3, 8, 18, 16, 12, 2, 7, 10, 15, 11, 10, 4, 4, 11, 15, 22

Offset: 0

Zhi-Wei Sun, Apr 10 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 2, 4, 5, 7, 9, 21, 22, 43. Also, every n = 0,1,2,... can be written as pen(u) + pen(v) + pen(x) + pen(y) + pen(z) with u,v,x,y,z nonnegative integers such that 3*u + 5*v + 11*x + 16*y + 19*z is also a pentagonal number.
(ii) Any integer n > 43 can be written as the sum of five pentagonal numbers u, v, x, y and z such that u + 2*v + 5*x + 7*y + 10*z is also a pentagonal number. Also, each integer n > 10 can be written as the sum of five pentagonal numbers u, v, x, y and z such that u + 2*v + 5*x + 7*y + 10*z is a square.
(iii) Any natural number n can be written as u^2 + v^2 + x^2 + y^2 + z^2 with u^2 + 2*v^2 + 3*x^2 + 4*y^2 + 5*z^2 a square, where u, v, x, y and z are integers.
As conjectured by Fermat and proved by Cauchy, each natural number can be written as the sum of five pentagonal numbers.
See also A271510, A271513, A271518 and A271644 for some similar conjectures refining Lagrange's four-square theorem.

			a(7) = 1 since 7 = 5 + 0 + 1 + 0 + 1 = pen(2) + pen(0) + pen(1) + pen(0) + pen(1) with 2 + 2*0 + 4*1 + 5*0 + 6*1 = 12 = pen(3).
a(9) = 1 since 9 = 1 + 1 + 5 + 1 + 1 = pen(1) + pen(1) + pen(2) + pen(1) + pen(1) with 1 + 2*1 + 4*2 + 5*1 + 6*1 = 22 = pen(4).
a(22) = 1 since 22 = 0 + 0 + 5 + 12 + 5 = pen(0) + pen(0) + pen(2) + pen(3) + pen(2) with 0 + 2*0 + 4*2 + 5*3 + 6*2 = 35 = pen(5).
a(43) = 1 since 43 = 5 + 1 + 35 + 1 + 1 = pen(2) + pen(1) + pen(5) + pen(1) + pen(1) with 2 + 2*1 + 4*5 + 5*1 + 6*1 = 35 = pen(5).

Zhi-Wei Sun, Table of n, a(n) for n = 0..1500
Z.-W. Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.

Cf. A000290, A000326, A271510, A271513, A271518, A271644.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
pen[x_]:=pen[x]=x*(3x-1)/2
pQ[n_]:=pQ[n]=SQ[24n+1]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0)
Do[r=0;Do[If[pQ[n-pen[x]-pen[y]-pen[z]-pen[w]]&&pQ[x+2y+4z+5w+6*Floor[(Sqrt[24(n-pen[x]-pen[y]-pen[z]-pen[w])+1]+1)/6]],r=r+1],{x,0,(Sqrt[24n+1]+1)/6},{y,0,(Sqrt[24(n-pen[x])+1]+1)/6},{z,0,(Sqrt[24(n-pen[x]-pen[y])+1]+1)/6},{w,0,(Sqrt[24(n-pen[x]-pen[y]-pen[z])+1]+1)/6}];Print[n," ",r];Label[aa];Continue,{n,0,70}]

A271714 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that (10w+5x)^2 + (12y+36z)^2 is a square, where w is a positive integer and x,y,z are nonnegative integers.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 3, 1, 2, 1, 2, 3, 1, 4, 4, 2, 2, 1, 3, 3, 5, 2, 2, 5, 2, 1, 2, 3, 3, 3, 2, 3, 2, 3, 4, 4, 2, 3, 9, 2, 3, 1, 1, 6, 2, 3, 4, 6, 4, 1, 2, 5, 3, 3, 4, 3, 5, 1, 4, 5, 1, 3, 6, 6, 1, 3, 4, 5, 12, 2, 4, 6, 2, 4

Offset: 1

Zhi-Wei Sun, Apr 12 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 7, 9, 19, 49, 133, 589, 2^k, 2^k*3, 4^k*q (k = 0,1,2,... and q = 14, 67, 71, 199).
(ii) If P(y,z) is one of 2y-3z, 2y-8z and 4y-6z, then any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that (w-x)^2 + P(y,z)^2 is a square.
(iii) For each triple (a,b,c) = (1,4,4), (1,12,12), (2,4,8), (2,6,6), (2,12,12), (3,4,4), (3,4,8), (3,8,8), (3,12,12), (3,12,36), (5,4,4), (5,4,8), (5,8,16), (5,36,36), (6,4,4), (7,12,12), (7,20,20), (7,24,24), (9,4,4), (9,12,12),(9,36,36), (11,12,12), (13,4,4), (15,12,12), (16,12,12), (21,20,20), (21,24,24), (23,12,12), any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that (w+a*x)^2 + (b*y-c*z)^2 is a square.
See also A271510, A271513, A271518, A271644, A271665, A271721 and A271724 for other conjectures refining Lagrange's four-square theorem.

			a(2) = 1 since 2 = 1^2 + 1^2 + 0^2 + 0^2 with (10*1+5*1)^2 + (12*0+36*0)^2 = 15^2 + 0^2 = 15^2.
a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with (10*1+5*1)^2 + (12*0+36*1)^2 = 15^2 + 36^2 = 39^2.
a(4) = 1 since 4 = 2^2 + 0^2 + 0^2 + 0^2 with (10*2+5*0)^2 + (12*0+36*0)^2 = 20^2 + 0^2 = 20^2.
a(6) = 1 since 6 = 2^2 + 0^2 + 1^2 + 1^2 with (10*2+5*0)^2 + (12*1+36*1)^2 = 20^2 + 48^2 = 52^2.
a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with (10*1+5*2)^2 + (12*1+36*1)^2 = 20^2 + 48^2 = 52^2.
a(9) = 1 since 9 = 3^2 + 0^2 + 0^2 + 0^2 with (10*3+5*0)^2 + (12*0+36*0)^2 = 30^2 + 0^2 = 30^2.
a(19) = 1 since 19 = 3^2 + 0^2 + 3^2 + 1^2 with (10*3+5*0)^2 + (12*3+36*1)^2 = 30^2 + 72^2 = 78^2.
a(49) = 1 since 49 = 7^2 + 0^2 + 0^2 + 0^2 with (10*7+5*0)^2 + (12*0+36*0)^2 = 70^2 + 0^2 = 70^2.
a(133) = 1 since 133 = 9^2 + 0^2 + 6^2 + 4^2 with (10*9+5*0)^2 + (12*6+36*4)^2 = 90^2 + 216^2 = 234^2.
a(589) = 1 since 589 = 17^2 + 10^2 + 2^2 + 14^2 with (10*17+5*10)^2 + (12*2+36*14)^2 = 220^2 + 528^2 = 572^2.

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

Cf. A000118, A000290, A271510, A271513, A271518, A271608, A271644, A271665, A271719, A271721, A271724.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(10*Sqrt[n-x^2-y^2-z^2]+5x)^2+(12y+36z)^2],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,0,Sqrt[n-1-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A271665 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that w^2 + 4xy + 8yz + 32zx is a square, where w is a positive integer and x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 12 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*3^m, 4^k*3^m*43, 4^k*9^m*q (k,m = 0, 1, 2, ... and q = 7, 15, 79, 95, 141, 159, 183).
(ii) Any positive integer n can be written as w^2 + x^2 + y^2 + z^2 with w*x + x*y + 2*y*z + 3*z*x (or w*x + 3*x*y + 8*y*z + 5*z*x) twice a square, where w is a positive integer and x,y,z are nonnegative integers.
(iii) For each k = 1, 2, 8, any positive integer can be written as w^2 + x^2 + y^2 + z^2 with w^2 + k*(x*y+y*z) a square, where w is a positive integer and x,y,z are nonnegative integers.
(iv) For each ordered pair (b,c) = (16,4), (24,4), (32,16), any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^2 + b*y^2 + c*x*z + c*y*z + c*z*w is a square.
We also guess that for each triple (b,c,d) = (1,3,4), (1,6,8), (1,7,24), (1,8,15), (1,10,24), (1,12,35), (1,14,48), (1,20,48), (2,1,2), (2,4,2), (2,4,7), (2,6,7), (2,8,4), (2,8,14), (2,8,31), (2,10,23), (2,12,14), (2,12,34), (2,14,47), (3,1,1), (3,1,4), (3,1,16), (3,2,14), (3,2,17), (3,2,38), (3,3,3), (3,3,13), (3,4,1), (3,4,4), (3,5,11), (3,6,3), (3,6,6), (3,6,26), (3,8,2), (3,8,13), (3,8,22), (3,9,39), (3,12,12), (3,12,33), (3,15,1), any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers and x^2 + b*y^2 + c*x*z + d*y*z a square.
See also A271510, A271513, A271518 and A271644 for other conjectures refining Lagrange's four-square theorem.

			a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 1^2 + 4*0*1 + 8*1*1 + 32*1*0 = 3^2.
a(4) = 1 since 4 = 2^2 + 0^2 + 0^2 + 0^2 with 2^2 + 4*2*0 + 8*0*0 + 32*0*0 = 2^2.
a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with 1^2 + 4*2*1 + 8*1*1 + 32*1*2 = 9^2.
a(15) = 1 since 15 = 1^2 + 2^2 + 1^2 + 3^2 with 1^2 + 4*2*1 + 8*1*3 + 32*3*2 = 15^2.
a(43) = 1 since 43 = 3^2 + 3^2 + 4^2 + 3^2 with 3^2 + 4*3*4 + 8*4*3 + 32*3*3 = 21^2.
a(79) = 1 since 79 = 5^2 + 3^2 + 6^2 + 3^2 with 5^2 + 4*3*6 + 8*6*3 + 32*3*3 = 23^2.
a(95) = 1 since 95 = 5^2 + 6^2 + 5^2 + 3^2 with 5^2 + 4*6*5 + 8*5*3 + 32*3*6 = 29^2.
a(129) = 1 since 129 = 5^2 + 6^2 + 8^2 + 2^2 with 5^2 + 4*6*8 + 8*8*2 + 32*2*6 = 27^2.
a(141) = 1 since 141 = 8^2 + 5^2 + 4^2 + 6^2 with 8^2 + 4*5*4 + 8*4*6 + 32*6*5 = 36^2.
a(159) = 1 since 159 = 11^2 + 1^2 + 6^2 + 1^2 with 11^2 + 4*1*6 + 8*6*1 + 32*1*1 = 15^2.
a(183) = 1 since 183 = 1^2 + 9^2 + 10^2 + 1^2 with 1^2 + 4*9*10 + 8*10*1 + 32*1*9 = 27^2.

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

Cf. A000118, A000290, A271510, A271513, A271518, A271608, A271644.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[4x*y+8*y*z+32*z*x+(n-x^2-y^2-z^2)],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,0,Sqrt[n-1-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A271724 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w(x+2y+3*z) a square, where w,x,y,z are nonnegative integers with x > 0.

Original entry on oeis.org

1, 3, 2, 1, 4, 4, 1, 3, 4, 6, 4, 2, 4, 7, 1, 1, 10, 8, 5, 6, 8, 5, 1, 4, 7, 10, 7, 2, 11, 13, 2, 3, 8, 9, 8, 6, 7, 13, 3, 6, 15, 8, 4, 4, 13, 8, 1, 2, 8, 15, 11, 4, 14, 18, 5, 7, 6, 6, 12, 5, 12, 17, 5, 1, 16, 21, 3, 11, 16, 12, 1, 8, 8, 18, 16, 5, 16, 12, 4, 6

Offset: 1

Zhi-Wei Sun, Apr 13 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 7, 15, 47, 151, 4^k*q (k = 0,1,2,... and q = 1, 23, 71).
(ii) For positive integers a,b,c with gcd(a,b,c) squarefree, any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers and w*(a*x+b*y+c*z) a square, if and only if {a,b,c} is among {1,2,3}, {1,3,6}, {1,6,9}, {5,6,9}, {18,30,114}.
(iii) For each quadruple (a,b,c,d) = (1,1,2,12), (1,2,7,60), (1,3,9,48), (1,4,11,48), (1,5,8,24), (1,8,11,24), (2,6,8,15), (3,5,6,24), (3,6,15,40), (3,6,18,40), (3,12,15,20), (4,4,8,15), (4,8,12,21), (4,8,12,45), (4,8,20,15), (4,8,36,45), (5,10,15,24), (6,9,15,20), (7,14,28,60), (7,21,28,60), (7,21,42,60), (12,36,48,55), (14,21,28,60), (3,9,18,112), (3,21,33,80), (4,5,9,120), (4,12,16,105), any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that (a*x+b*y+c*z)^2 + (d*w)^2 is a square.
See also A271510, A271513, A271518, A271644, A271665, A271714 and A271721 for other conjectures refining Lagrange's four-square theorem.

			a(1) = 1 since 1 = 0^2 + 1^2 + 0^2 + 0^2 with 1 > 0 and 0*(1+2*0+3*0) = 0^2.
a(3) = 2 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1*(1+2*0+3*1) = 2^2, and 3 = 0^2 + 1^2 + 1^2 + 1^2 with 0*(1+2*1+3*1) = 0^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1*(1+2*1+3*2) = 3^2.
a(15) = 1 since 15 = 2^2 + 3^2 + 1^2 + 1^2 with 2*(3+2*1+3*1) = 4^2.
a(23) = 1 since 23 = 1^2 + 3^2 + 2^2 + 3^2 with 1*(3+2*2+3*3) = 4^2.
a(31) = 2 since 31 = 2^2 + 1^2 + 1^2 + 5^2 with 2*(1+2*1+3*5) = 6^2, and also 31 = 2^2 + 3^2 + 3^2 + 3^2 with 2*(3+2*3+3*3) = 6^2.
a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1*(1+2*3+3*6) = 5^2.
a(71) = 1 since 71 = 1^2 + 6^2 + 5^2 + 3^2 with 1*(6+2*5+3*3) = 5^2.
a(151) = 1 since 151 = 9^2 + 6^2 + 5^2 + 3^2 with 9*(6+2*5+3*3) = 15^2.

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

Cf. A000118, A000290, A259789, A271510, A271513, A271518, A271608, A271644, A271665, A271714, A271721.

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+2y+3z)],r=r+1],{x,1,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Label[aa];Continue,{n,1,80}]

A271775 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 (x >= y >= z <= w) with x - y a square, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Apr 13 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 3, 11, 47, 2^{4k+3}*m (k = 0,1,2,... and m = 1, 3, 7, 15, 79).
(ii) Let a and b be positive integers with a <= b and gcd(a,b) squarefree. Then any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x-b*y a square, if and only if (a,b) is among the ordered pairs (1,1), (2,1), (2,2), (4,3), (6,2). Verified for all nonnegative integers up to 10^11. - Mauro Fiorentini, Jun 14 2024
(iii) Let a and b be positive integers with gcd(a,b) squarefree. Then any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x+b*y a square, if and only if {a,b} is among {1,2}, {1,3} and {1,24}. Verified for all nonnegative integers up to 10^11. - Mauro Fiorentini, Jun 14 2024
(iv) Let a,b,c be positive integers with a <= b and gcd(a,b,c) squarefree. Then, any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x+b*y-c*z a square, if and only if (a,b,c) is among the triples (1,1,1), (1,1,2), (1,2,1), (1,2,2), (1,2,3), (1,3,1), (1,3,3), (1,4,4), (1,5,1), (1,6,6), (1,8,6), (1,12,4), (1,16,1), (1,17,1), (1,18,1), (2,2,2), (2,2,4), (2,3,2), (2,3,3), (2,4,1), (2,4,2), (2,6,1), (2,6,2), (2,6,6), (2,7,4), (2,7,7), (2,8,2), (2,9,2), (2,32,2), (3,3,3), (3,4,2), (3,4,3), (3,8,3), (4,5,4), (4,8,3), (4,9,4), (4,14,14), (5,8,5), (6,8,6), (6,10,8), (7,9,7), (7,18,7), (7,18,12), (8,9,8), (8,14,14), (8,18,8), (14,32,14), (16,18,16), (30,32,30), (31,32,31), (48,49,48), (48,121,48). Verified for all nonnegative integers up to 10^11. - Mauro Fiorentini, Jun 14 2024
(v) Let a,b,c be positive integers with b <= c and gcd(a,b,c) squarefree. Then, any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x-b*y-c*z a square, if and only if (a,b,c) is among the triples (1,1,1), (2,1,1), (2,1,2), (3,1,2) and (4,1,2).
(vi) Let a,b,c,d be positive integers with a <= b, c <= d and gcd(a,b,c,d) squarefree. Then, any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x+b*y-(c*z+d*w) a square, if and only if (a,b,c,d) is among the quadruples (1,2,1,1), (1,2,1,2), (1,3,1,2), (1,4,1,3), (2,4,1,2), (2,4,2,4), (8,16,7,8), (9,11,2,9) and (9,16,2,7).
(vii) Let a,b,c,d be positive integers with a <= b <= c and gcd(a,b,c,d) squarefree. Then, any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x+b*y+c*z-d*w a square, if and only if (a,b,c,d) is among the quadruples (1,1,2,1), (1,2,3,1), (1,2,3,3), (1,2,4,2), (1,2,4,4), (1,2,5,5), (1,2,6,2), (1,2,8,1), (2,2,4,4), (2,4,6,4), (2,4,6,6), and (2,4,8,2).
It is known that any natural number not of the form 4^k*(16*m+14) (k,m = 0,1,2,...) can be written as x^2 + y^2 + 2*z^2 = x^2 + y^2 + z^2 + z^2 with x,y,z nonnegative integers.
See also A271510, A271513, A271518, A271644, A271665, A271714, A271721 and A271724 for other conjectures refining Lagrange's four-square theorem.

			a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1 = 1 > 0 < 1 and 1 - 1 = 0^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 = 1 = 1 < 2 and 1 - 1 = 0^2.
a(8) = 1 since 8 = 2^2 + 2^2 + 0^2 + 0^2 with 2 = 2 > 0 = 0 and 2 - 2 = 0^2.
a(11) = 1 since 11 = 1^2 + 1^2 + 0^2 + 3^2 with 1 = 1 > 0 < 3 and 1 - 1 = 0^2.
a(24) = 1 since 24 = 2^2 + 2^2 + 0^2 + 4^2 with 2 = 2 > 0 < 4 and 2 - 2 = 0^2.
a(47) = 1 since 47 = 3^2 + 3^2 + 2^2 + 5^2 with 3 = 3 > 2 < 5 and 3 - 3 = 0^2.
a(53) = 2 since 53 = 3^2 + 2^2 + 2^2 + 6^2 with 3 > 2 = 2 < 6 and 3 - 2 = 1^2, and also 53 = 6^2 + 2^2 + 2^2 + 3^2 with 6 > 2 = 2 < 3 and 6 - 2 = 2^2.
a(56) = 1 since 56 = 6^2 + 2^2 + 0^2 + 4^2 with 6 > 2 > 0 < 4 and 6 - 2 = 2^2.
a(120) = 1 since 120 = 8^2 + 4^2 + 2^2 + 6^2 with 8 > 4 > 2 < 6 and 8 - 4 = 2^2.
a(632) = 1 since 632 = 16^2 + 12^2 + 6^2 + 14^2 with 16 > 12 > 6 < 14 and 16 - 12 = 2^2.

L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939, pp. 112-113.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015; Sci. China Math. 58(2015), 1367-1396.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A259789, A271510, A271513, A271518, A271608, A271644, A271665, A271714, A271721, A271724.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[x-y]&&SQ[n-x^2-y^2-z^2],r=r+1],{z,0,Sqrt[n/4]},{y,z,Sqrt[(n-z^2)/2]},{x,y,Sqrt[(n-y^2-z^2)]}];Print[n," ",r];Continue,{n,0,70}]

A271721 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= z >= 0, x > 0 and w >= z such that (x-y)*(w-z) is a square.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 12 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 5, 11, 15, 23, 35, 95, 4^k*190 (k = 0,1,2,...).
(ii) For each k = 4, 5, 6, 7, 8, 11, 12, 13, 15, 17, 18, 20, 22, 25, 27, 29, 33, 37, 38, 41, 50, 61, any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that (x-y)*(w-k*z) is a square.
(iii) For each triple (a,b,c) = (3,1,1), (1,2,1), (2,2,1), (3,2,1), (2,2,2), (6,2,1), (1,3,1), (3,3,1), (15,3,1), (1,4,1), (2,4,1), (1,5,1), (3,5,1), (5,5,1), (1,5,2), (1,6,1), (2,6,1), (3,6,1), (15,6,1), (1,7,1), (5,7,1), (1,8,1), (1,8,5), (3,9,1), (1,10,1), (1,12,1), (1,13,1), (3,13,1), (1,14,1), (1,15,1), (1,15,2), (6,16,1), (2,18,1), (3,18,1), (1,20,2), (1,21,1), (3,21,1), (1,23,1), (1,24,1), (1,27,1), (3,27,1), (1,34,1), (1,45,1), (3,45,1), (3,48,1), (1,55,1), (1,60,1), (5,60,1), 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)*(w-c*z) is a square.
This is stronger than Lagrange's four-square theorem. Note that for k = 2 or 3, any natural number n can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers and (x-y)*(w-k*z) = 0, for, if n cannot be represented by x^2 + y^2 + 2*z^2 then it has the form 4^k*(16*m+14) (k,m = 0,1,2,...) and hence it can be represented by x^2 + y^2 + (k^2+1)*z^2. It is known that natural numbers not represented by x^2 + y^2 + 5*z^2 have the form 4^k*(8*m+3), and that positive even numbers not represented by x^2 + y^2 + 10*z^2 have the form 4^k*(16*m+6) (as conjectured by S. Ramanujan and proved by L. E. Dickson).
See also A271510, A271513, A271518, A271644, A271714 and A271724 for other conjectures refining Lagrange's theorem.

			a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 0^2 with 1 = 1 > 0 = 0 and (1-1)*(0-0) = 0^2.
a(5) = 1 since 5 = 2^2 + 1^2 + 0^2 + 0^2 with 2 > 1 > 0 = 0 and (2-1)*(0-0) = 0^2.
a(11) = 1 since 11 = 1^2 + 1^2 + 0^2 + 3^2 with 1 = 1 > 0 < 3 and (1-1)*(3-0) = 0^2.
a(14) = 2 since 14 = 3^2 + 1^2 + 0^2 + 2^2 with 3 > 1 > 0 < 2 and (3-1)*(2-0) = 2^2, and also 14 = 3^2 + 2^2 + 0^2 + 1^2 with 3 > 2 > 0 < 1 and (3-2)*(1-0) = 1^2.
a(15) = 1 since 15 = 3^2 + 2^2 + 1^2 + 1^2 with 3 > 2 > 1 = 1 and (3-2)*(1-1) = 0^2.
a(23) = 1 since 23 = 3^2 + 3^2 + 1^2 + 2^2 with 3 = 3 > 1 < 2 and (3-3)*(2-1) = 0^2.
a(35) = 1 since 35 = 3^2 + 3^2 + 1^2 + 4^2 with 3 = 3 > 1 < 4 and (3-3)*(4-1) = 0^2.
a(95) = 1 since 95 = 5^2 + 5^2 + 3^2 + 6^2 with 5 = 5 > 3 < 6 and (5-5)*(6-3) = 0^2.
a(190) = 1 since 190 = 13^2 + 4^2 + 1^2 + 2^2 with 13 > 4 > 1 < 2 and (13-4)*(2-1) = 3^2.

L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33(1927), 63-70.
L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939, pp. 112-113.

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

Cf. A000118, A000290, A259789, A271510, A271513, A271518, A271608, A271644, A271665, A271714, A271719, A271724.

  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]-z)*(x-y)],r=r+1],{z,0,Sqrt[n/4]},{y,z,Sqrt[(n-z^2)/2]},{x,Max[1,y],Sqrt[(n-y^2-2z^2)]}];Print[n," ",r];Continue,{n,1,100}]

A271778 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and x^2 + 3y^2 + 5z^2 - 8*w^2 a square.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Apr 14 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 15, 29, 33, 47, 53, 65, 89, 129, 689, 1553, 2^(2k+1)*m (k = 0,1,2,... and m = 1, 29).
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x^2 + b*y^2 + c*z^2 - d*w^2 a square, if (a,b,c,d) is among the following quadruples: (1,3,6,3), (1,3,9,3), (1,3,30,3), (1,4,12,4), (1,4,20,4), (1,5,20,5), (1,5,35,20), (3,4,9,3), (3,9,40,3), (4,5,16,4), (4,11,33,11), (4,12,16,7), (5,16,20,20), (5,25,36,5), (6,10,25,10), (9,12,28,12), (9,21,28,21), (15,21,25,15), (15,24,25,15), (1,5,60,5), (1,20,60,20), (9, 28,63,63), (9,28,84,84), (12,33,64,12), (16,21,105,21), (16,33,64,16), (21,25,45,45), (24,25,75,75), (24,25,96,96), (25,40,96,40), (25,48,96,48), (25,60,84,60), (25,60,96,60), (25,75,126,75), (32,64,105,32).
(iii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x^2 + b*y^2 - c*z^2 -d*w^2 a square, whenever (a,b,c,d) is among the quadruples (3,9,3,20), (5,9,5,20), (5,25,4,5), (9,81,9,20),(12,16,3,12), (16,64,15,16), (20,25,4,20), (27,81,20,27), (30,64,15,30), (32,64,15,32), (48,64,15,48), (60,64,15,60), (60,81,20,60), (64,80,15,80).
(iv) For each triple (a,b,c) = (21,5,15), (36,3,8), (48,8,39), (64,7,8), (40,15,144), (45,20,144), (69,20,60), any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x^2 - b*y^2 - c*z^2 a square.
See also A271510, A271513, A271518, A271665, A271714, A271721, A271724 and A271775 for other conjectures refining Lagrange's four-square theorem.

			a(2) = 1 since 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1^2 + 3*1^2 + 5*0^2 - 8*0^2 = 2^2.
a(15) = 1 since 15 = 1^2 + 3^2 + 1^2 + 2^2 with 1^2 + 3*3^2 + 5*1^2 - 8*2^2 = 1^2.
a(29) = 1 since 29 = 3^2 + 4^2 + 0^2 + 2^2 with 3^2 + 3*4^2 + 5*0^2 - 8*2^2 = 5^2.
a(33) = 1 since 33 = 2^2 + 4^2 + 2^2 + 3^2 with 2^2 + 3*4^2 + 5*2^2 - 8*3^2 = 0.
a(47) = 1 since 47 = 5^2 + 3^2 + 2^2 + 3^2 with 5^2 + 3*3^2 + 5*2^2 - 8*3^2 = 0^2.
a(53) = 1 since 53 = 3^2 + 2^2 + 6^2 + 2^2 with 3^2 + 3*2^2 + 5*6^2 - 8*2^2 = 13^2.
a(58) = 1 since 58 = 4^2 + 1^2 + 5^2 + 4^2 with 4^2 + 3*1^2 + 5*5^2 - 8*4^2 = 4^2.
a(65) = 1 since 65 = 3^2 + 6^2 + 2^2 + 4^2 with 3^2 + 3*6^2 + 5*2^2 - 8*4^2 = 3^2.
a(89) = 1 since 89 = 6^2 + 4^2 + 6^2 + 1^2 with 6^2 + 3*4^2 + 5*6^2 - 8*1^2 = 16^2.
a(129) = 1 since 129 = 9^2 + 4^2 + 4^2 + 4^2 with 9^2 + 3*4^2 + 5*4^2 - 8*4^2 = 9^2.
a(689) = 1 since 689 = 11^2 + 18^2 + 10^2 + 12^2 with 11^2 + 3*18^2 + 5*10^2 - 8*12^2 = 21^2.
a(1553) = 1 since 1553 = 21^2 + 6^2 + 26^2 + 20^2 with 21^2 + 3*6^2 + 5*26^2 - 8*20^2 = 27^2.

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, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x^2+3*y^2+5*z^2-8*(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}]

A271824 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+2y)^2 + 8z^2 + 40*w^2 a square, where x is a positive integer and y,z,w are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 14 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 9, 11, 15, 23, 33, 71, 129, 167, 187, 473, 4^k*m (k = 0,1,2,... and m = 1, 22, 38, 278). Also, any positive integer can be written as x^2 + y^2 + z^2 + w^2 with 9*(x+2*y)^2 + 16*z^2 + 24*w^2 a square, where x is a positive integer and y,z,w are nonnegative integers.
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and (a*x-b*y)^2 + c*z^2 + d*w^2 a square, provided that (a,b,c,d) is among the quadruples (4,8,1,8), (12,24,1,24), (2,4,5,40), (3,6,7,9), (3,9,7,9), (3,6,7,63), (1,2,8,16), (1,2,8,40), (3,6,8,40), (2,6,9,12), (3,5,9,15), (4,8,9,16), (12,24,9,16), (3,6,15,25), (3,6,16,24), (3,12,16,24), (6,9,16,24), (9,12,16,24), (4,8,16,41), (8,12,16,41), (3,6,16,48), (6,9,16,48), (2,3,16,56), (3,6,28,63), (2,4,36,45), (6,12,40,45), (7,14,56,64) and (2,6,57,60).
(iii) Let a and be positive integers with a <= b and gcd(a,b) squarefree. Then any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and (a*x+b*y)*z a square, if and only if (a,b) is among the ordered pairs (1,1), (1,2), (1,3), (2,5), (3,3), (3,6), (3,15), (5,6), (5,11), (5,13), (6,15), (8,46) and (9,23).
(iv) Let a and b be positive integers with a <= b and gcd(a,b) squarefree. Then, any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and (a*x^2+b*y^2)*z a square, if and only if (a,b) is among the ordered pairs (3,13), (5,11), (15,57), (15,165) and (138,150).
There are many ordered pairs (a,b) of integers with gcd(a,b) squarefree such that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers and a*x^2 + b*y^2 a square. For example, we have shown that (1,-1), (2,-2), (3,-3) and (1,2) are indeed such ordered pairs.
See also A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775 and A271778 for other conjectures refining Lagrange's four-square theorem.

			a(9) = 1 since 9 = 3^2 + 0^2 + 0^2 + 0^2 with (3+2*0)^2 + 8*0^2 + 40*02 = 3^2.
a(11) = 1 since 11 = 1^2 + 1^2 + 3^2 + 0^2 with (1+2*1)^2 + 8*3^2 + 40*0^2 = 9^2.
a(15) = 1 since 15 = 1^2 + 3^2 + 2^2 + 1^2 with (1+2*3)^2 + 8*2^2 + 40*1^2 = 11^2.
a(22) = 1 since 22 = 3^2 + 2^2 + 3^2 + 0^2 with (3+2*2)^2 + 8*3^2 + 40*0^2 = 11^2.
a(23) = 1 since 23 = 1^2 + 3^2 + 2^2 + 3^2 with (1+2*3)^2 + 8*2^2 + 40*3^2 = 21^2.
a(33) = 1 since 33 = 4^2 + 1^2 + 0^2 + 4^2 with (4+2*1)^2 + 8*0^2 + 40*4^2 = 26^2.
a(38) = 1 since 38 = 5^2 + 2^2 + 0^2 + 3^2 with (5+2*2)^2 + 8*0^2 + 40*3^2 = 21^2.
a(71) = 1 since 71 = 1^2 + 6^2 + 5^2 + 3^2 since (1+2*6)^2 + 8*5^2 + 40*3^2 = 27^2.
a(129) = 1 since 129 = 5^2 + 6^2 + 8^2 + 2^2 with (5+2*6)^2 + 8*8^2 + 40*2^2 = 31^2.
a(167) = 1 since 167 = 11^2 + 1^2 + 3^2 + 6^2 with (11+2*1)^2 + 8*3^2 + 40*6^2 = 41^2.
a(187) = 1 since 187 = 3^2 + 5^2 + 12^2 + 3^2 with (3+2*5)^2 + 8*12^2 + 40*3^2 = 41^2.
a(278) = 1 since 278 = 3^2 + 0^2 + 10^2 + 13^2 with (3+2*0)^2 + 8*10^2 + 40*13^2 = 87^2.
a(473) = 1 since 473 = 7^2 + 10^2 + 0^2 + 18^2 with (7+2*10)^2 + 8*0^2 + 40*18^2 = 117^2.

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

Cf. A000118, A000290, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+2y)^2+8z^2+40(n-x^2-y^2-z^2)],r=r+1],{x,1,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

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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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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A271608 Number of ordered ways to write n as pen(u) + pen(v) + pen(x) + pen(y) + pen(z) with u,v,x,y,z nonnegative integers such that u + 2*v + 4*x + 5*y + 6*z is a pentagonal number, where pen(k) denotes the pentagonal number k*(3k-1)/2.

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A271714 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that (10*w+5*x)^2 + (12*y+36*z)^2 is a square, where w is a positive integer and x,y,z are nonnegative integers.

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A271665 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that w^2 + 4*x*y + 8*y*z + 32*z*x is a square, where w is a positive integer and x,y,z are nonnegative integers.

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A271724 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w*(x+2*y+3*z) a square, where w,x,y,z are nonnegative integers with x > 0.

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A271775 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 (x >= y >= z <= w) with x - y a square, where w,x,y,z are nonnegative integers.

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A271721 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= z >= 0, x > 0 and w >= z such that (x-y)*(w-z) is a square.

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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.

A271608 Number of ordered ways to write n as pen(u) + pen(v) + pen(x) + pen(y) + pen(z) with u,v,x,y,z nonnegative integers such that u + 2v + 4x + 5y + 6z is a pentagonal number, where pen(k) denotes the pentagonal number k*(3k-1)/2.

A271714 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that (10w+5x)^2 + (12y+36z)^2 is a square, where w is a positive integer and x,y,z are nonnegative integers.

A271665 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that w^2 + 4xy + 8yz + 32zx is a square, where w is a positive integer and x,y,z are nonnegative integers.

A271724 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w(x+2y+3*z) a square, where w,x,y,z are nonnegative integers with x > 0.

A271778 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and x^2 + 3y^2 + 5z^2 - 8*w^2 a square.

A271824 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+2y)^2 + 8z^2 + 40*w^2 a square, where x is a positive integer and y,z,w are nonnegative integers.