A000118 -id:A000118 - cp's OEIS frontend

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

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

Offset: 1

Zhi-Wei Sun, May 08 2016

nonn

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

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

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

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

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

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

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

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

A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 4, 6, 1, 0, 2, 0, 12, 8, 1, 0, 0, 4, 8, 24, 10, 1, 0, 0, 8, 6, 32, 40, 12, 1, 0, 0, 0, 24, 24, 80, 60, 14, 1, 0, 0, 0, 24, 48, 90, 160, 84, 16, 1, 0, 2, 4, 0, 96, 112, 252, 280, 112, 18, 1, 0, 0, 4, 12, 64, 240, 312, 574, 448, 144, 20, 1

Offset: 0

Peter Luschny, Oct 01 2018

Number of ways of writing k as a sum of n squares.

			[ 0] 1,  0,    0,    0,     0,     0,     0      0,     0,     0, ... A000007
[ 1] 1,  2,    0,    0,     2,     0,     0,     0,     0,     2, ... A000122
[ 2] 1,  4,    4,    0,     4,     8,     0,     0,     4,     4, ... A004018
[ 3] 1,  6,   12,    8,     6,    24,    24,     0,    12,    30, ... A005875
[ 4] 1,  8,   24,   32,    24,    48,    96,    64,    24,   104, ... A000118
[ 5] 1, 10,   40,   80,    90,   112,   240,   320,   200,   250, ... A000132
[ 6] 1, 12,   60,  160,   252,   312,   544,   960,  1020,   876, ... A000141
[ 7] 1, 14,   84,  280,   574,   840,  1288,  2368,  3444,  3542, ... A008451
[ 8] 1, 16,  112,  448,  1136,  2016,  3136,  5504,  9328, 12112, ... A000143
[ 9] 1, 18,  144,  672,  2034,  4320,  7392, 12672, 22608, 34802, ... A008452
[10] 1, 20,  180,  960,  3380,  8424, 16320, 28800, 52020, 88660, ... A000144
   A005843,   v, A130809,  v,  A319576,  v ,   ...      diagonal: A066535
           A046092,    A319575,       A319577,     ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954.
J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006).

Seiichi Manyama, Descending antidiagonals n = 0..139, flattened
L. Carlitz, Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124.
S. H. Chan, An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Index entries for sequences related to sums of squares

Variant starting with row 1 is A122141, transpose of A286815.

A319574row := proc(n, len) series(JacobiTheta3(0, x)^n, x, len+1);
[seq(coeff(%, x, j), j=0..len-1)] end:
seq(print([n], A319574row(n, 10)), n=0..10);
# Alternative, uses function PMatrix from A357368.
PMatrix(10, n -> A000122(n-1)); # Peter Luschny, Oct 19 2022

A[n_, k_] := If[n == k == 0, 1, SquaresR[n, k]];
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

for n in (0..10):
    Q = DiagonalQuadraticForm(ZZ, [1]*n)
    print(Q.theta_series(10).list())

A046895 Sizes of successive clusters in Z^4 lattice.

Original entry on oeis.org

1, 9, 33, 65, 89, 137, 233, 297, 321, 425, 569, 665, 761, 873, 1065, 1257, 1281, 1425, 1737, 1897, 2041, 2297, 2585, 2777, 2873, 3121, 3457, 3777, 3969, 4209, 4785, 5041, 5065, 5449, 5881, 6265, 6577, 6881, 7361, 7809, 7953, 8289, 9057

Offset: 0

N. J. A. Sloane

Number of lattice points inside or on the 4-sphere x^2 + y^2 + z^2 + u^2 = n. - T. D. Noe, Mar 14 2009

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms up to 1000 from Noe)
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.

Partial sums of A000118.

Cf. A117609

Accumulate[ Table[ SquaresR[4, n], {n, 0, 42}]] (* Jean-François Alcover, May 11 2012 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]^2*QP[q^4]^2))^4/(1-q) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Joerg Arndt *)

q='q+O('q^66);
Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4/(1-q))
/* Joerg Arndt, Apr 08 2013 */

from math import isqrt
def A046895(n): return 1+((-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=n>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # Chai Wah Wu, Jun 21 2024

a(n) = A122510(4,n). a(n^2) = A055410(n). - R. J. Mathar, Apr 21 2010
G.f.: T3(q)^4/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - Joerg Arndt, Apr 08 2013
Pi^2/2 * (sqrt(n)-1)^4 < a(n) < Pi^2/2 * (sqrt(n)+1)^4 for n > 0. - Charles R Greathouse IV, Feb 17 2015
a(n) = Pi^2/2 * n^2 + O(n (log n)^(2/3)) using a result of Walfisz. - Charles R Greathouse IV, Feb 18 2015
a(n) = 1 + 8*A024916(n) - 32*A024916(floor(n/4)) by Jacobi's four-square theorem. - Peter J. Taylor, Jun 03 2020

A096727 Expansion of eta(q)^8 / eta(q^2)^4 in powers of q.

Original entry on oeis.org

1, -8, 24, -32, 24, -48, 96, -64, 24, -104, 144, -96, 96, -112, 192, -192, 24, -144, 312, -160, 144, -256, 288, -192, 96, -248, 336, -320, 192, -240, 576, -256, 24, -384, 432, -384, 312, -304, 480, -448, 144, -336, 768, -352, 288, -624, 576, -384, 96, -456, 744, -576, 336, -432, 960, -576, 192

Offset: 0

Michael Somos, Jul 06 2004

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 8*q + 24*q^2 - 32*q^3 + 24*q^4 - 48*q^5 + 96*q^6 - 64*q^7 + 24*q^8 - ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.

Cf. A000118, A002131, A004011, A005879, A109506.

# JacobiTheta4 is defined in A002448.
A096727List(len) = JacobiTheta4(len, 4)
A096727List(57) |> println # Peter Luschny, Mar 12 2018

A := Basis( ModularForms( Gamma0(4), 2), 57); A[1] - 8*A[2]; /* Michael Somos, Aug 21 2014 */

CoefficientList[ Series[1 + Sum[k(-8x^k/(1 - x^k) + 48x^(2k)/(1 - x^(2k)) - 64x^(4k)/(1 - x^(4k))), {k, 1, 60}], {x, 0, 60}], x] (* Robert G. Wilson v, Jul 14 2004 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ q Dt[ Log @ m, q], {q, 0, n}]]; (* Michael Somos, Sep 06 2012 *)
a[ n_] := (-1)^n SquaresR[ 4, n]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)
QP = QPochhammer; s = QP[q]^8/QP[q^2]^4 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)

{a(n) = if( n<1, n==0, 8 * (-1)^n * sumdiv( n, d, if( d%4, d)))};

{a(n) = local(A); if( n<0, 0, A = x *O (x^n); polcoeff( eta(x + A)^8 / eta(x^2 + A)^4, n))};

A = ModularForms( Gamma0(4), 2, prec=57) . basis(); A[0] - 8*A[1]; # Michael Somos, Jun 12 2014

a(n) =  -8*sigma(n) + 48*sigma(n/2) - 64*sigma(n/4) for n>0, where sigma(n) = A000203(n) if n is an integer, otherwise 0.
Euler transform of period 2 sequence [ -8, -4, ...].
G.f.: Prod_{k>0} (1 - x^k)^8 / (1 - x^(2k))^4 = 1 + Sum_{k>0} k * (-8 * x^k / (1 - x^k) + 48 * x^(2*k)  /(1 - x^(2*k)) - 64 * x^(4*k)/(1 - x^(4*k))).
G.f. theta_4(q)^4 = (Sum_{k} (-q)^(k^2))^4.
Expansion of phi(-q)^4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 01 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = v^4 - 30*u*v^2*w + 12*u*v*w * (u + 9*w) - u*w * (u^2 + 9*w*u + 81*w^2).
a(n) = (-1)^n * A000118(n). a(n) = 8 * A109506(n) unless n=0. a(2*n) = A004011(n). a(2*n + 1) = -A005879(n).
a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017

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

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 11 2016

nonn

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

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

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

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

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

A279612 Number of ways to write n = x^2 + y^2 + z^2 + w^2 with x + 2y - 2z a power of 4 (including 4^0 = 1), where x,y,z,w are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 15 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 16^k*q (k = 0,1,2,... and q = 1, 2, 3, 6, 7, 8, 12, 15, 27, 31, 47, 72, 76, 92, 111, 127).
(ii) Let a and b be positive integers with gcd(a,b) odd. Then any positive integer 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 power of two (including 2^0 = 1) if and only if (a,b) = (1,1), (2,1), (2,3).
(iii) Let a,b,c be positive integers with a <= b and gcd(a,b,c) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y - c*z a power of two if and only if (a,b,c) is among the triples (1,1,1), (1,1,2), (1,2,1), (1,2,2), (1,3,1), (1,3,2), (1,3,3), (1,3,4), (1,3,5), (1,4,1), (1,4,2), (1,4,3), (1,4,4), (1,5,1), (1,5,2), (1,5,4), (1,5,5,), (1,6,3), (1,7,4), (1,7,7), (1,8,1), (1,9,2), (2,3,1), (2,3,3), (2,3,4), (2,5,1), (2,5,3), (2,5,4), (2,5,5), (2,7,1), (2,7,3), (2,7,7), (2,9,3), (2,11,5), (3,4,3), (7,8,7).
(iv) Let a,b,c be positive integers with b <= c and gcd(a,b,c) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x - b*y - c*z a power of two if and only if (a,b,c) is among the triples (2,2,1), (4,2,1), (4,3,1), (4,4,3).
(v) Let a,b,c be positive integers with a <= b, c <= d, and gcd(a,b,c,d) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y - c*z -d*w a power of two if and only if (a,b,c,d) is among the quadruples (1,2,1,1), (1,2,1,2), (1,2,1,3), (1,3,1,2), (1,3,2,3), (1,3,2,4), (1,4,1,2), (1,7,2,6), (1,9,1,4), (2,2,2,3), (2,3,1,2), (2,3,1,3), (2,3,2,3), (2,3,6,1), (2,4,1,2), (2,5,1,2), (2,5,2,3), (2,5,3,4), (3,4,1,2), (3,4,1,3), (3,4,1,5), (3,4,2,5), (3,4,3,4), (3,8,1,10), (3,8,2,3), (4,5,1,5).
(vi) Let a,b,c be positive integers with a <= b <= c and gcd(a,b,c,d) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y + c*z -d*w a power of two if and only if (a,b,c,d) is among the quadruples (1,1,2,2), (1,1,2,3), (1,1,2,4), (1,2,2,3), (1,2,3,4), (1,2,4,3), (1,2,6,7), (1,3,4,4), (1,4,6,5), (2,3,5,4).
(vii) For any positive integers a,b,c,d, not all positive integers can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x - b*y - c*z -d*w a power of two.
(viii) Let a and b be positive integers, and c and d be nonnegative integers. Then, not all positive integers can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y + c*z + d*w a power of two.
We have verified a(n) > 0 for all n = 1..2*10^7. The conjecture that a(n) > 0 for all n > 0 appeared in arXiv:1701.05868.

			a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 + 2*1 - 2*1 = 4^0.
a(15) = 1 since 15 = 3^2 + 1^2 + 2^2 + 1^2 with 3 + 2*1 - 2*2 = 4^0.
a(27) = 1 since 27 = 4^2 + 1^2 + 1^2 + 3^2 with 4 + 2*1 - 2*1 = 4.
a(31) = 1 since 31 = 3^2 + 2^2 + 3^2 + 3^2 with 3 + 2*2 - 2*3 = 4^0.
a(47) = 1 since 47 = 3^2 + 2^2 + 3^2 + 5^2 with 3 + 2*2 - 2*3 = 4^0.
a(72) = 1 since 72 = 8^2 + 0^2 + 2^2 + 2^2 with 8 + 2*0 - 2*2 = 4.
a(76) = 1 since 76 = 1^2 + 5^2 + 5^2 + 5^2 with 1 + 2*5 - 2*5 = 4^0.
a(92) = 1 since 92 = 4^2 + 6^2 + 6^2 + 2^2 with 4 + 2*6 - 2*6 = 4.
a(111) = 1 since 111 = 9^2 + 1^2 + 5^2 + 2^2 with 9 + 2*1 - 2*5 = 4^0.
a(127) = 1 since 127 = 7^2 + 2^2 + 5^2 + 7^2 with 7 + 2*2 - 2*5 = 4^0.

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. Also available from arXiv:1604.06723 [math.NT].
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000079, A000118, A000290, A271518, A275656, A275675, A275676, A275738, A278560, A279616.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
FP[n_]:=FP[n]=n>0&&IntegerQ[Log[4,n]];
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&FP[x+2y-2z],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,1,80}]

A300441 Number of the integers 4^k(4u(m)^2+1) (k,m = 0,1,2,...) such that n^2 - 4^k(4u(m)^2+1) can be written as the sum of two squares, where u(0) = 0, u(1) = 1, and u(j+1) = 4*u(j) - u(j-1) for j = 1,2,3,....

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 05 2018

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 2^k*m (k = 0,1,2,... and m = 1, 7).
This curious conjecture indicates that any positive square can be written as (2^k)^2 + (2^(k+1)*u(m))^2 + x^2 + y^2 with k,m,x,y nonnegative integers. In the 2017 JNT paper, the author proved that each n = 1,2,3,... can be written as 4^k*(1+4*x^2+y^2)+z^2 with k,x,y,z nonnegative integers.
We have verified that a(n) > 0 for all n = 1..10^7.

			a(1) = 1 since 1^2 - 4^0*(4*u(0)^2+1) = 1 is 1^2 + 0^2.
a(5) = 3 since 5^2 - 4^0*(4*u(1)^2+1) = 20 = 4^2 + 2^2, 5^2 - 4^1*(4*u(1)^2+1) = 5 = 2^2 + 1^2, and 5^2 - 4^2*(4*u(0)^2+1) = 9 = 3^2 + 0^2.
a(7) = 1 since 7^2 - 4^1*(4*u(0)^2+1) = 45 = 6^2 + 3^2.

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], 2017-2018.

Cf. A000118, A000290, A000302, A001353, A001481, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396.

u[0]=0;
u[1]=1;
u[n_]:=u[n]=4u[n-1]-u[n-2];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[If[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1,1,0],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
tab={};Do[r=0;Do[m=0;Label[cc];If[4u[m]^2+1>n^2/4^k,Goto[bb]];If[QQ[n^2-4^k*(4u[m]^2+1)],r=r+1,m=m+1;Goto[cc]];
Label[bb],{k,0,Log[2,n]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A236924 Number of integer solutions to a^2 + 2b^2 + 2c^2 + 4*d^2 = n.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 16, 16, 24, 26, 24, 24, 32, 28, 32, 48, 24, 36, 52, 40, 48, 64, 48, 48, 96, 62, 56, 80, 64, 60, 96, 64, 24, 96, 72, 96, 104, 76, 80, 112, 144, 84, 128, 88, 96, 156, 96, 96, 96, 114, 124, 144, 112, 108, 160, 144, 192, 160, 120, 120, 192, 124, 128, 208, 24, 168, 192, 136, 144, 192, 192

Offset: 0

N. J. A. Sloane, Feb 14 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Olivia X. M. Yao, Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622.

Cf. A000118, A320149, A320150, A320151, A320152, A320153.

with(LinearAlgebra);
L:=proc(a,b,c,d,M) local n,hit,ul,vl,wl,xl,u,v,w,x,su,sv,sw,sx;
hit:=Array(0..M); ul:=floor(sqrt(M/a)); vl:=floor(sqrt(M/b));
wl:=floor(sqrt(M/c)); xl:=floor(sqrt(M/d));
for u from -ul to ul do
su:=a*u^2;
for v from -vl to vl do
sv:=su+b*v^2;
for w from -wl to wl do
sw:=sv+c*w^2;
for x from -xl to xl do
sx:=sw+d*x^2;
if sx <= M then hit[sx]:=hit[sx]+1; fi;
od; od; od; od;
[seq(hit[n],n=0..M)];
end;
L(1,2,2,4,60);

CoefficientList[EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^2]^2*EllipticTheta[3, 0, q^4] + O[q]^100, q] (* Jean-François Alcover, Feb 15 2023, after Ilya Gutkovskiy *)

Yao and Xia give an explicit formula for a(n) due to Liouville.

G.f.: theta_3(q)*theta_3(q^2)^2*theta_3(q^4), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 03 2018

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

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 13 2016

nonn

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 15 2016

nonn

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

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

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

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

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

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

cp's OEIS Frontend

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

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A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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A046895 Sizes of successive clusters in Z^4 lattice.

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A096727 Expansion of eta(q)^8 / eta(q^2)^4 in powers of q.

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Julia

Magma

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

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A279612 Number of ways to write n = x^2 + y^2 + z^2 + w^2 with x + 2*y - 2*z a power of 4 (including 4^0 = 1), where x,y,z,w are nonnegative integers.

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A300441 Number of the integers 4^k*(4*u(m)^2+1) (k,m = 0,1,2,...) such that n^2 - 4^k*(4*u(m)^2+1) can be written as the sum of two squares, where u(0) = 0, u(1) = 1, and u(j+1) = 4*u(j) - u(j-1) for j = 1,2,3,....

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

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

A279612 Number of ways to write n = x^2 + y^2 + z^2 + w^2 with x + 2y - 2z a power of 4 (including 4^0 = 1), where x,y,z,w are nonnegative integers.

A300441 Number of the integers 4^k(4u(m)^2+1) (k,m = 0,1,2,...) such that n^2 - 4^k(4u(m)^2+1) can be written as the sum of two squares, where u(0) = 0, u(1) = 1, and u(j+1) = 4*u(j) - u(j-1) for j = 1,2,3,....

A236924 Number of integer solutions to a^2 + 2b^2 + 2c^2 + 4*d^2 = n.

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