cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A275344 Number of ordered ways to write n as x^2 + y^2 + z^2 + 2*w^2 with x + 2*y + 3*z a square, where x,y,z,w are nonnegative integers.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 1, 1, 3, 5, 3, 4, 3, 3, 2, 4, 1, 4, 3, 3, 4, 1, 4, 3, 1, 4, 3, 3, 8, 3, 2, 3, 2, 3, 2, 3, 3, 3, 4, 2, 2, 9, 3, 8, 7, 5, 5, 4, 2, 6, 4, 4, 9, 4, 4, 5, 4, 3, 8, 6, 5, 6, 5, 5, 5, 4, 2, 5, 5, 4, 6, 4
Offset: 0

Views

Author

Zhi-Wei Sun, Jul 24 2016

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 3, 5, 7, 14, 15, 16, 25, 30, 33, 84, 169, 225.
(ii) For each ordered pair (a,b) = (1,2), (1,3), (1,12), (1,23), (2,3), (2,4), (2,6), (2,7), (2,15), (2,16), any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that a*x + b*y is a square.
This is similar to the conjecture in A271518. It is known that any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers.

Examples

			a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 2*0^2 with 0 + 2*0 + 3*0 = 0^2.
a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 2*0^2 with 1 + 2*0 + 3*0 = 1^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 0^2 + 2*1^2 with 1 + 2*0 + 3*0 = 1^2.
a(5) = 1 since 5 = 2^2 + 1^2 + 0^2 + 2*0^2 with 2 + 2*1 + 3*0 = 2^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 0^2 + 2*1^2 with 2 + 2*1 + 3*0 = 2^2.
a(14) = 1 since 14 = 1^2 + 1^2 + 2^2 + 2*2^2 with 1 + 2*1 + 3*2 = 3^2.
a(15) = 1 since 15 = 3^2 + 0^2 + 2^2 + 2*1^2 with 3 + 2*0 + 3*2 = 3^2.
a(16) = 1 since 16 = 4^2 + 0^2 + 0^2 + 2*0^2 with 4 + 2*0 + 3*0 = 2^2.
a(25) = 1 since 25 = 1^2 + 4^2 + 0^2 + 2*2^2 with 1 + 2*4 + 3*0 = 3^2.
a(30) = 1 since 30 = 3^2 + 2^2 + 3^2 + 2*2^2 with 3 + 2*2 + 3*3 = 4^2.
a(33) = 1 since 33 = 1^2 + 0^2 + 0^2 + 2*4^2 with 1 + 2*0 + 3*0 = 1^2.
a(84) = 1 since 84 = 4^2 + 6^2 + 0^2 + 2*4^2 with 4 + 2*6 + 3*0 = 4^2.
a(169) = 1 since 169 = 10^2 + 6^2 + 1^2 + 2*4^2 with 10 + 2*6 + 3*1 = 5^2.
a(225) = 1 since 225 = 10^2 + 6^2 + 9^2 + 2*2^2 with 10 + 2*6 + 3*9 = 7^2.

Links

Crossrefs

Cf. A000290, A271518, A275297, A275301.

Programs

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

A275298 Number of ordered ways to write n as w^3 + x^2 + y^2 + z^2 with x - w a square, where x,y,z,w are nonnegative integers with y <= z > w.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jul 22 2016

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 4, 7, 8, 12, 16, 23, 24, 40, 47, 71, 167, 311, 599.
(ii) For each triple (a,b,c) = (1,1,1), (2,1,1), (2,1,2), (2,2,2), (3,1,2), any natural number can be written as x^2 + y^2 + z^2 + w^3 with x,y,z,w nonnegative integers such that a*y - b*z - c*w is a square.
See also A275297 and A275299 for similar conjectures.

Examples

			a(1) = 1 since 1 = 0^3 + 0^2 + 0^2 + 1^2 with 0 - 0 = 0^2 and 0 < 1 > 0.
a(3) = 1 since 3 = 0^3 + 1^2 + 1^2 + 1^2 with 1 - 0 = 1^2 and 1 = 1 > 0.
a(4) = 1 since 4 = 0^3 + 0^2 + 0^2 + 2^2 with 0 - 0 = 0^2 and 0 < 2 > 0.
a(7) = 1 since 7 = 1^3 + 1^2 + 1^2 + 2^2 with 1 - 1 = 0^2 and 1 < 2 > 1.
a(8) = 1 since 8 = 0^3 + 0^2 + 2^2 + 2^2 with 0 - 0 = 0^2 and 2 = 2 > 0.
a(12) = 1 since 12 = 1^3 + 1^2 + 1^2 + 3^2 with 1 - 1 = 0^2 and 1 < 3 > 1.
a(16) = 1 since 16 = 0^3 + 0^2 + 0^2 + 4^2 with 0 - 0 = 0^2 and 0 < 4 > 0.
a(23) = 1 since 23 = 1^3 + 2^2 + 3^2 + 3^2 with 2 - 1 = 1^2 and 3 = 3 > 1.
a(24) = 1 since 24 = 0^3 + 4^2 + 2^2 + 2^2 with 4 - 0 = 2^2 and 2 = 2 > 0.
a(40) = 1 since 40 = 0^3 + 0^2 + 2^2 + 6^2 with 0 - 0 = 0^2 and 2 < 6 > 0.
a(47) = 1 since 47 = 1^3 + 1^2 + 3^2 + 6^2 with 1 - 1 = 0^2 and 3 < 6 > 1.
a(71) = 1 since 71 = 1^3 + 5^2 + 3^2 + 6^2 with 5 - 1 = 2^2 and 3 < 6 > 1.
a(167) = 1 since 167 = 1^3 + 2^2 + 9^2 + 9^2 with 2 - 1 = 1^2 and 9 = 9 > 1.
a(311) = 1 since 311 = 1^3 + 2^2 + 9^2 + 15^2 with 2 - 1 = 1^2 and 9 < 15 > 1.
a(599) = 1 since 599 = 5^3 + 5^2 + 7^2 + 20^2 with 5 - 5 = 0^2 and 7 < 20 > 5.

Links

Crossrefs

Cf. A000290, A000578, A271518, A275297, A275299.

Programs

  • Mathematica
    SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[r=0;Do[If[CQ[n-x^2-y^2-z^2]&&SQ[x-(n-x^2-y^2-z^2)^(1/3)]&&(n-x^2-y^2-z^2)^(1/3)

A275299 Number of ordered ways to write n as x^3 + y^2 + z^2 + w^2 with x + y - z a square, where x,y,z,w are nonnegative integers with y <= z.

Original entry on oeis.org

1, 2, 3, 3, 2, 1, 3, 2, 1, 4, 4, 2, 3, 4, 2, 2, 2, 3, 7, 4, 1, 2, 5, 2, 1, 4, 2, 5, 4, 2, 3, 2, 3, 4, 6, 3, 3, 8, 4, 2, 2, 2, 6, 4, 4, 4, 4, 2, 4, 5, 4, 8, 3, 4, 3, 3, 3, 4, 7, 1, 4, 4, 5, 3, 4, 5, 5, 8, 3, 5, 5, 1, 7, 10, 3, 3, 6, 5, 5, 1, 5
Offset: 0

Views

Author

Zhi-Wei Sun, Jul 22 2016

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 5, 8, 20, 24, 59, 71, 79, 119, 184, 575, 743, 764, 1471, 2759.
(ii) For each triple (a,b,c) = (1,2,1), (3,1,1), (4,2,2), (4,3,3), (5,1,1), (6,2,2), (14,2,2), any natural number can be written as x^3 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that a*x + b*y - c*z is a square.
See also A275297 and A275298 for similar conjectures.

Examples

			a(1) = 2 since 1 = 0^3 + 0^2 + 0^2 + 1^2 with 0 + 0 - 0 = 0^2 and 0 = 0, and also 1 = 1^3 + 0^2 + 0^2 + 0^2 with 1 + 0 - 0 = 1^2 and 0 = 0.
a(5) = 1 since 5 = 1^3 + 0^2 + 0^2 + 2^2 with 1 + 0 - 0 = 1^2 and 0 = 0.
a(8) = 1 since 8 = 0^3 + 2^2 + 2^2 + 0^2 with 0 + 2 - 2 = 0^2 and 2 = 2.
a(20) = 1 since 20 = 1^3 + 3^2 + 3^2 + 1^2 with 1 + 3 - 3 = 1^2 and 3 = 3.
a(24) = 1 since 24 = 0^3 + 2^2 + 2^2 + 4^2 with 0 + 2 - 2 = 0^2 and 2 = 2.
a(59) = 1 since 59 = 0^3 + 5^2 + 5^2 + 3^2 with 0 + 5 - 5 = 0^2 and 5 = 5.
a(71) = 1 since 71 = 1^3 + 5^2 + 6^2 + 3^2 with 1 + 5 - 6 = 0^2 and 5 < 6.
a(79) = 1 since 79 = 3^3 + 4^2 + 6^2 + 0^2 with 3 + 4 - 6 = 1^2 and 4 < 6.
a(119) = 1 since 119 = 1^3 + 3^2 + 3^2 + 10^2 with 1 + 3 - 3 = 1^2 and 3 = 3.
a(184) = 1 since 184 = 5^3 + 3^2 + 7^2 + 1^2 with 5 + 3 - 7 = 1^2 and 3 < 7.
a(575) = 1 since 575 = 7^3 + 0^2 + 6^2 + 14^2 with 7 + 0 - 6 = 1^2 and 0 < 6.
a(743) = 1 since 743 = 1^3 + 2^2 + 3^2 + 27^2 with 1 + 2 - 3 = 0^2 and 2 < 3.
a(764) = 1 since 764 = 7^3 + 9^2 + 12^2 + 14^2 with 7 + 9 - 12 = 2^2 and 9 < 12.
a(1471) = 1 since 1471 = 1^3 + 25^2 + 26^2 + 13^2 with 1 + 25 - 26 = 0^2 and 25 < 26.
a(2759) = 1 since 2759 = 5^3 + 8^2 + 13^2 + 49^2 with 5 + 8 - 13 = 0^2 and 8 < 13.

Links

Crossrefs

Cf. A000290, A000578, A271518, A275297, A275298.

Programs

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

A275301 Number of ordered ways to write n as x^2 + y^2 + z^2 + 2*w^2 with x + 2*y a cube, where x,y,z,w are nonnegative integers.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 3, 4, 4, 5, 3, 2, 5, 2, 4, 5, 2, 3, 4, 3, 1, 3, 4, 4, 5, 3, 3, 5, 6, 3, 4, 3, 2, 4, 3, 4, 4, 3, 3, 7, 3, 4, 5, 3, 6, 4, 4, 4, 3, 3, 2, 3, 2, 2, 8, 3, 4, 8, 4, 3, 8, 3, 4, 9, 3, 4, 3, 4, 1, 3, 4
Offset: 0

Views

Author

Zhi-Wei Sun, Jul 22 2016

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 5, 6, 7, 8, 11, 14, 15, 30, 78, 90, 93, 106, 111, 117, 125, 223, 335.
(ii) Any natural number can be written as x^2 + y^2 + z^2 + 2*w^3 with x,y,z,w nonnegative integers such that x + 2*y is a square.
See also A275344 for a similar conjecture.

Examples

			a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 2*0^2 with 0 + 2*0 = 0^3.
a(2) = 2 since 2 = 1^2 + 0^2 + 1^2 + 2*0^2 with 1 + 2*0 =1^3, and 2 = 0^2 + 0^2 + 0^2 + 2*1^2 with 0 + 2*0 = 0^3.
a(5) = 1 since 5 = 1^2 + 0^2 + 2^2 + 2*0^2 with 1 + 2*0 = 1^3.
a(6) = 1 since 6 = 0^2 + 0^2 + 2^2 + 2*1^2 with 0 + 2*0 = 0^3.
a(7) = 1 since 7 = 1^2 + 0^2 + 2^2 + 2*1^2 with 1 + 2*0 = 1^3.
a(8) = 1 since 8 = 0^2 + 0^2 + 0^2 + 2*2^2 with 0 + 2*0 = 0^3.
a(11) = 1 since 11 = 0^2 + 0^2 + 3^2 + 2*1^2 with 0 + 2*0 = 0^3.
a(14) = 1 since 14 = 2^2 + 3^2 + 1^2 + 2*0^2 with 2 + 2*3 = 2^3.
a(15) = 1 since 15 = 2^2 + 3^2 + 0^2 + 2*1^2 with 2 + 2*3 = 2^3.
a(30) = 1 since 30 = 2^2 + 3^2 + 3^2 + 2*2^2 with 2 + 2*3 = 2^3.
a(78) = 1 since 78 = 6^2 + 1^2 + 3^2 + 2*4^2 with 6 + 2*1 = 2^3.
a(90) = 1 since 90 = 1^2 + 0^2 + 9^2 + 2*2^2 with 1 + 2*0 = 1^3.
a(93) = 1 since 93 = 4^2 + 2^2 + 1^2 + 2*6^2 with 4 + 2*2 = 2^3.
a(106) = 1 since 106 = 4^2 + 2^2 + 6^2 + 2*5^2 with 4 + 2*2 = 2^3.
a(111) = 1 since 111 = 2^2 + 3^2 + 0^2 + 2*7^2 with 2 + 2*3 = 2^3.
a(117) = 1 since 117 = 4^2 + 2^2 + 5^2 + 2*6^2 with 4 + 2*2 = 2^3.
a(125) = 1 since 125 = 6^2 + 1^2 + 4^2 + 2*6^2 with 6 + 2*1 = 2^3.
a(223) = 1 since 223 = 11^2 + 8^2 + 6^2 + 2*1^2 with 11 + 2*8 = 3^3.
a(335) = 1 since 335 = 11^2 + 8^2 + 10^2 + 2*5^2 with 11 + 2*8 = 3^3.

Links

Crossrefs

Cf. A000290, A000578, A271518, A275297, A275300, A275344.

Programs

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

A275300 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + y + z a square, where x,y,z are integers with x >= |y| <= |z|, and w is a nonnegative integer.

Original entry on oeis.org

1, 3, 3, 3, 2, 1, 5, 4, 3, 5, 4, 5, 1, 2, 9, 4, 4, 4, 7, 6, 1, 2, 6, 1, 7, 7, 8, 6, 3, 5, 7, 1, 7, 11, 11, 9, 4, 5, 6, 4, 3, 15, 10, 8, 2, 7, 9, 1, 4, 9, 5, 12, 5, 11, 10, 3, 8, 5, 3, 8, 7, 10, 10, 2, 4, 11, 9, 8, 6, 10, 13, 1, 7, 10, 8, 8, 2, 10, 14, 3, 10
Offset: 0

Views

Author

Zhi-Wei Sun, Jul 22 2016

Keywords

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 5, 12, 20, 23, 31, 47, 71, 103, 148, 164.
The author proved in arXiv:1604.06723 that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x + y + z a square, where x,y,z,w are integers.
See also A275297, A275298, A275299 and A272620 for similar conjectures.

Examples

			a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^3 with 0 + 0 + 0 = 0^2 and 0 = 0 = 0.
a(5) = 1 since 5 = 2^2 + 0^2 + (-1)^2 + 0^3 with 2 + 0 + (-1) = 1^2 and 2 > 0 < |-1|.
a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^3 with 3 + (-1) + (-1) = 1^2 and 3 > |-1| = |-1|.
a(20) = 1 since 20 = 3^2 + 1^2 + (-3)^2 + 1^3 with 3 + 1 + (-3) = 1^2 and 3 > 1 < |-3|.
a(23) = 1 since 23 = 3^2 + (-2)^2 + 3^2 + 1^3 with 3 + (-2) + 3 = 2^2 and 3 > |-2| < 3.
a(31) = 1 since 31 = 5^2 + 1^2 + (-2)^2 + 1^3 with 5 + 1 + (-2) = 2^2 and 5 > 1 < |-2|.
a(47) = 1 since 47 = 6^2 + 1^2 + (-3)^2 + 1^3 with 6 + 1 + (-3) = 2^2 and 6 > 1 < |-3|.
a(71) = 1 since 71 = 6^2 + 3^2 + (-5)^2 + 1^3 with 6 + 3 + (-5) = 2^2 and 6 > 3 < |-5|.
a(103) = 1 since 103 = 7^2 + 2^2 + 7^2 + 1^3 with 7 + 2 + 7 = 4^2 and 7 > 2 < 7.
a(148) = 1 since 148 = 9^2 + (-2)^2 + (-6)^2 + 3^3 with 9 + (-2) + (-6) = 1^2 and 9 > |-2| < |-6|.
a(164) = 1 since 164 = 9^2 + 1^2 + (-9)^2 + 1^3 with 9 + 1 + (-9) = 1^2 and 9 > 1 < |-9|.

Links

Crossrefs

Cf. A000290, A000578, A271518, A272620, A275297, A275298, A275299.

Programs

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

A275409 Number of ordered ways to write n as 2*w^2 + x^2 + y^2 + z^2 with w + x + 2*y + 4*z a square, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

1, 2, 1, 0, 2, 2, 2, 1, 1, 1, 0, 3, 1, 2, 1, 1, 3, 2, 5, 3, 4, 3, 1, 1, 1, 1, 2, 2, 2, 4, 2, 2, 4, 2, 7, 3, 1, 6, 2, 1, 2, 3, 4, 5, 1, 1, 3, 5, 3, 3, 4, 3, 7, 3, 2, 4, 3, 4, 4, 3, 1, 4, 5, 3, 6, 4, 4, 4, 5, 7, 7, 3, 6, 5, 5, 4, 3, 11, 2, 2, 4
Offset: 0

Views

Author

Zhi-Wei Sun, Jul 26 2016

Keywords

Comments

Conjecture: (i) a(n) > 0 except for n = 3, 10, and a(n) = 1 only for n = 0, 2, 7, 8, 9, 12, 14, 15, 22, 23, 24, 25, 36, 39, 44, 45, 60, 87, 98, 106, 110, 111, 183.
(ii) Any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that x + 2*y + 3*z - 3*w is a square.
(iii) For each triple (a,b,c) = (1,2,1), (1,2,3), (1,3,1), (2,4,1), (2,4,2), (2,4,3), (2,4,4), (2,4,8), (8,9,5), any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that a*x + b*y - c*z is a square.
(iv) Any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that x + 2*y - 2*z is twice a nonnegative cube. Also, each natural number can be written as x^2 + y^2 + z^2 + 2*w^3 with x,y,z,w nonnegative integers such that x + 3*y - z is a square.
See also A275344 and A275301 for related conjectures. We are able to show that each natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w integers such that x + y + z = t^2 for some t = 0, 1, 2.

Examples

			a(2) = 1 since 2 = 2*1^2 + 0^2 + 0^2 + 0^2 with 1 + 0 + 2*0 + 4*0 = 1^2.
a(7) = 1 since 7 = 2*1^2 + 0^2 + 2^2 + 1^2 with 1 + 0 + 2*2 + 4*1 = 3^2.
a(8) = 1 since 8 = 2*1^2 + 2^2 + 1^2 + 1^2 with 1 + 2 + 2*1 + 4*1 = 3^2.
a(9) = 1 since 9 = 2*2^2 + 0^2 + 1^2 + 0^2 with 2 + 0 + 2*1 + 4*0 = 2^2.
a(12) = 1 since 12 = 2*2^2 + 2^2 + 0^2 + 0^2 with 2 + 2 + 2*0 + 4*0 = 2^2.
a(14) = 1 since 14 = 2*0^2 + 2^2 + 1^2 + 3^2 with 0 + 2 + 2*1 + 4*3 = 4^2.
a(15) = 1 since 15 = 2*1^2 + 2^2 + 3^2 + 0^2 with 1 + 2 + 2*3 + 4*0 = 3^2.
a(22) = 1 since 22 = 2*1^2 + 4^2 + 2^2 + 0^2 with 1 + 4 + 2*2 + 4*0 = 3^2.
a(23) = 1 since 23 = 2*3^2 + 2^2 + 0^2 + 1^2 with 3 + 2 + 2*0 + 4*1 = 3^2.
a(24) = 1 since 24 = 2*0^2 + 4^2 + 2^2 + 2^2 with 0 + 4 + 2*2 + 4*2 = 4^2.
a(25) = 1 since 25 = 2*0^2 + 4^2 + 0^2 + 3^2 with 0 + 4 + 2*0 + 4*3 = 4^2.
a(36) = 1 since 36 = 2*3^2 + 1^2 + 4^2 + 1^2 with 3 + 1 + 2*4 + 4*1 = 4^2.
a(39) = 1 since 39 = 2*1^2 + 6^2 + 1^2 + 0^2 with 1 + 6 + 2*1 + 4*0 = 3^2.
a(44) = 1 since 44 = 2*3^2 + 0^2 + 1^2 + 5^2 with 3 + 0 + 2*1 + 4*5 = 5^2.
a(45) = 1 since 45 = 2*0^2 + 5^2 + 2^2 + 4^2 with 0 + 5 + 2*2 + 4*4 = 5^2.
a(60) = 1 since 60 = 2*2^2 + 6^2 + 4^2 + 0^2 with 2 + 6 + 2*4 + 4*0 = 4^2.
a(87) = 1 since 87 = 2*3^2 + 2^2 + 8^2 + 1^2 with 3 + 2 + 2*8 + 4*1 = 5^2.
a(98) = 1 since 98 = 2*4^2 + 1^2 + 8^2 + 1^2 with 4 + 1 + 2*8 + 4*1 = 5^2.
a(106) = 1 since 106 = 2*2^2 + 8^2 + 3^2 + 5^2 with 2 + 8 + 2*3 + 4*5 = 6^2.
a(110) = 1 since 110 = 2*6^2 + 5^2 + 3^2 + 2^2 with 6 + 5 + 2*3 + 4*2 = 5^2.
a(111) = 1 since 111 = 2*5^2 + 3^2 + 6^2 + 4^2 with 5 + 3 + 2*6 + 4*4 = 6^2.
a(183) = 1 since 183 = 2*3^2 + 10^2 + 4^2 + 7^2 with 3 + 10 + 2*4 + 4*7 = 7^2.

Links

Crossrefs

Cf. A000290, A271518, A275297, A275301, A275344.

Programs

  • Mathematica
    SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-2*w^2-x^2-y^2]&&SQ[w+x+2y+4*Sqrt[n-2*w^2-x^2-y^2]],r=r+1],{w,0,Sqrt[n/2]},{x,0,Sqrt[n-2*w^2]},{y,0,Sqrt[n-2*w^2-x^2]}];Print[n," ",r];Continue,{n,0,80}]
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