A000118 -id:A000118 - cp's OEIS frontend

A260625 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+3y+13z)xy*z a square, where x is a positive integer, and y,z,w are nonnegative integers with y >= z.

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

Offset: 1

Zhi-Wei Sun, Apr 30 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 7, 39, 47, 95, 191, 239, 327, 439, 871, 1167, 1199, 1367, 1487, 1727, 1751, 2063, 2351, 2471, 4647, 4^k*m (k = 0,1,2,... and m = 1, 3).
(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+c*z)*x*y*z is a square, whenever (a,b,c) is among the triples (1,3,7), (1,5,7), (1,5,11), (1,13,23), (2,4,6), (2,4,8), (2,6,8), (2,8,26), (3,5,21), (3,7,15), (3,9,43), (3,9,69), (3,9,141), (3,21,27), (3,27,39), (3,33,45), (3,39,123), (6,8,12), (6,8,18), (6,8,22), (6,8,28), (6,12,48), (6,18,132), (6,24,34), (6,24,36), (6,42,72), (7,13,29), (7,19,23), (12,18,24), (12,18,30), (12,26,48), (13,15,21), (13,17,19), (13,33,39), (14,28,58), (15,45,51), (16,22,62), (18,22,24), (21,27,33), (21,27,57), (23,37,61), (24,54,66), (33,57,79), (38,48,66), (42,58,84), (46,92,118).
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1 > 0 and (1+3*1+13*0)*1*1*0 =0^2.
a(4) = 1 since 4 = 2^2 + 0^2 + 0^2 + 0^2 with 2 > 0, 0 = 0 and (2+3*0+13*0)*2*0*0 = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2 > 0, 1 = 1 and
(2+3*1+13*1)*2*1*1 = 6^2.
a(39) = 1 since 39 = 2^2 + 3^2 + 1^2 + 5^2 with 2 > 0, 3 > 1 and (2+3*3+13*1)*2*3*1 = 12^2.
a(47) = 1 since 47 = 2^2 + 3^2 + 3^2 + 5^2 with 2 > 0, 3 = 3 and (2+3*3+13*3)*2*3*3 = 30^2.
a(95) = 1 since 95 = 2^2 + 3^2 + 1^2 + 9^2 with 2 > 0, 3 > 1 and (2+3*3+13*1)*2*3*1 = 12^2.
a(191) = 1 since 191 = 2^2 + 3^2 + 3^2 + 13^2 with 2 > 0, 3 = 3 and (2+3*3+13*3)*2*3*3 = 30^2.
a(239) = 1 since 239 = 2^2 + 3^2 + 1^2 + 15^2 with 2 > 0, 3 > 1 and (2+3*3+13*1)*2*3*1 = 12^2.
a(327) = 1 since 327 = 11^2 + 3^2 + 1^2 + 14^2 with 11 > 0, 3 > 1 and (11+3*3+13*1)*11*3*1 = 33^2.
a(439) = 1 since 439 = 10^2 + 5^2 + 5^2 + 17^2 with 10 > 0, 5 = 5 and (10+3*5+13*5)*10*5*5 = 150^2.
a(871) = 1 since 871 = 21^2 + 15^2 + 3^2 + 14^2 with 21 > 0, 15 > 3 and (21+3*15+13*3)*21*15*3 = 315^2.
a(1167) = 1 since 1167 = 22^2 + 11^2 + 11^2 + 21^2 with 22 > 0, 11 = 11 and (22+3*11+13*11)*22*11*11 = 726^2.
a(1199) = 1 since 1199 = 14^2 + 21^2 + 21^2 + 11^2 with 14 > 0, 21 = 21 and (14+3*21+13*21)*14*21*21 = 1470^2.
a(1367) = 1 since 1367 = 14^2 + 21^2 + 21^2 + 17^2 with 14 > 0, 21 = 21 and (14+3*21+13*21)*14*21*21 = 1470^2.
a(1487) = 1 since 1487 = 9^2 + 29^2 + 6^2 + 23^2 with 9 > 0, 29 > 6 and (9+3*29+13*6)*9*29*6 = 522^2.
a(1727) = 1 since 1727 = 2^2 + 21^2 + 21^2 + 29^2 with 2 > 0, 21 = 21 and (2+3*21+13*21)*2*21*21 = 546^2.
a(1751) = 1 since 1751 = 9^2 + 17^2 + 15^2 + 34^2 with 9 > 0, 17 > 15 and (9+3*17+13*15)*9*17*15 = 765^2.
a(2063) = 1 since 2063 = 18^2 + 19^2 + 3^2 + 37^2 with 18 > 0, 19 > 3 and (18+3*19+13*3)*18*19*3 = 342^2.
a(2351) = 1 since 2351 = 15^2 + 35^2 + 15^2 + 26^2 with 15 > 0, 35 > 15 and (15+3*35+13*15)*15*35*15 = 1575^2.
a(2471) = 1 since 2471 = 1^2 + 18^2 + 11^2 + 45^2 with 1 > 0, 18 > 11 and (1+3*18+13*11)*1*18*11 = 198^2.
a(4647) = 1 since 4647 = 10^2 + 45^2 + 29^2 + 41^2 with 10 > 0, 45 > 29 and (10+3*45+13*29)*10*45*29 = 2610^2.

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, A262357, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+3y+13z)x*y*z], r=r+1],{x,1,Sqrt[n]},{z,0,Sqrt[(n-x^2)/2]},{y,z,Sqrt[n-x^2-z^2]}];Print[n," ",r];Label[aa];Continue,{n,1,80}]

A272620 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w + x + y - z a square, where w is an integer and x,y,z are nonnegative integers with |w| <= x >= y <= z < x + y.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 03 2016

nonn

Conjecture: a(n) > 0 for all n > 0.
In contrast, the author has proved that any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that x + y + z is a square. See arXiv:1604.06723.
Yu-Chen Sun and the author proved in arXiv:1605.03074 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. - Zhi-Wei Sun, May 10 2016

			a(1) = 1 since 1 = 0^2 + 1^2 + 0^2 + 0^2 with 0 < 1 > 0 = 0 < 1 + 0 and 0 + 1 + 0 - 0 = 1^2.
a(2) = 1 since 2 = (-1)^2 + 1^2 + 0^2 + 0^2 with 1 = 1 > 0 = 0 < 1 + 0 and -1 + 1 + 0 - 0 = 0^2.
a(3) = 1 since 3 = 0^2 + 1^2 + 1^2 + 1^2 with 0 < 1 = 1 = 1 < 1 + 1 and 0 + 1 + 1 - 1 = 1^2.
a(4) = 1 since 4 = (-1)^2 + 1^2 + 1^2 + 1^2 with 1 = 1 = 1 = 1 < 1 + 1 and -1 + 1 + 1 - 1 = 0^2.
a(6) = 1 since 6 = (-1)^2 + 2^2 + 0^2 + 1^2 with 1 < 2 > 0 < 1 < 2 + 0 and -1 + 2 + 0 - 1 = 0^2.
a(7) = 1 since 7 = (-1)^2 + 2^2 + 1^2 + 1^2 with 1 < 2 > 1 = 1 < 2 + 1 and -1 + 2 + 1 - 1 = 1^2.
a(9) = 1 since 9 = 0^2 + 2^2 + 1^2 + 2^2 with 0 < 2 > 1 < 2 < 2 + 1 and 0 + 2 + 1 - 2 = 1^2.
a(11) = 1 since 11 = (-1)^2 + 3^2 + 0^2 + 1^2 with 1 < 3 > 0 < 1 < 3 + 0 and -1 + 3 + 0 - 1 = 1^2.
a(12) = 1 since 12 = 1^2 + 3^2 + 1^2 + 1^2 with 1 < 3 > 1 = 1 < 3 + 1 and 1 + 3 + 1 - 1 = 2^2.
a(17) = 1 since 17 = 0^2 + 2^2 + 2^2 + 3^2 with 0 < 2 = 2 < 3 < 2 + 2 and 0 + 2 + 2 - 3 = 1^2.
a(19) = 1 since 19 = 0^2 + 3^2 + 1^2 + 3^2 with 0 < 3 > 1 < 3 < 3 + 1 and 0 + 3 + 1 - 3 = 1^2.
a(23) = 1 since 23 = (-1)^2 + 3^2 + 2^2 + 3^2 with 1 < 3 > 2 < 3 < 3 + 2 and -1 + 3 + 2 - 3 = 1^2.
a(29) = 1 since 29 = 0^2 + 3^2 + 2^2 + 4^2 with 0 < 3 > 2 < 4 < 3 + 2 and 0 + 3 + 2 - 4 = 1^2.
a(31) = 1 since 31 = (-2)^2 + 3^2 + 3^2 + 3^2 with 2 < 3 = 3 = 3 < 3 + 3 and -2 + 3 + 3 - 3 = 1^2.
a(37) = 1 since 37 = (-1)^2 + 4^2 + 2^2 + 4^2 with 1 < 4 > 2 < 4 < 4 + 2 and -1 + 4 + 2 - 4 = 1^2.
a(92) = 1 since 92 = 3^2 + 5^2 + 3^2 + 7^2 with 3 < 5 > 3 < 7 < 5 + 3 and 3 + 5 + 3 - 7 = 2^2.
a(284) = 1 since 284 = 3^2 + 9^2 + 5^2 + 13^2 with 3 < 9 > 5 < 13 < 9 + 5 and 3 + 9 + 5 - 13 = 2^2.
a(572) = 1 since 572 = 3^2 + 11^2 + 9^2 + 19^2 with 3 < 11 > 9 < 19 < 11 + 9 and 3 + 11 + 9 - 19 = 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Yu-Chen Sun and Zhi-Wei Sun, Two refinements of Lagrange's four-square theorem, arXiv:1605.03074 [math.NT], 2016.
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, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.

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

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

A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.

Original entry on oeis.org

1, 24, 24, 96, 24, 144, 96, 192, 24, 312, 144, 288, 96, 336, 192, 576, 24, 432, 312, 480, 144, 768, 288, 576, 96, 744, 336, 960, 192, 720, 576, 768, 24, 1152, 432, 1152, 312, 912, 480, 1344, 144, 1008, 768, 1056, 288, 1872, 576, 1152, 96, 1368, 744, 1728, 336

Offset: 0

N. J. A. Sloane

D_4 is also the Barnes-Wall lattice in 4 dimensions.
E_{gamma,2} is the unique normalized modular form for Gamma_0(2) of weight 2.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
Convolution square is A008658. - Michael Somos, Aug 21 2014
Expansion of 2*P(x^2) - P(x) in powers of x where P() is a Ramanujan Eisenstein series. - Michael Somos, Feb 16 2015
a(n) is the number of Hurwitz quaternions of norm n. - Michael Somos, Feb 16 2015

			G.f. = 1 + 24*x + 24*x^2 + 96*x^3 + 24*x^4 + 144*x^5 + 96*x^6 + 192*x^7 + 24*x^8 + ...
G.f. = 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + 192*q^14 + 24*q^16 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1998, see p. 148 Eq. (9.11).
Harvey Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. Eq. (1).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 214.
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, and Sindy Xin Zhang, Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions, Williams College (2023).
Barry Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, Vol. 7, No. 3 (1998), 257-274.
Michael Gilleland, Some Self-Similar Integer Sequences.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
Nadia Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
Gabriele Nebe and N. J. A. Sloane, Home page for D_4 lattice
N. J. A. Sloane, The 24 minimal vectors form the 24-cell polytope.
N. J. A. Sloane, Seven Staggering Sequences.
Eric Weisstein's World of Mathematics, 24-Cell.
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice.
Eric Weisstein's World of Mathematics, Eisenstein Series.
Wikipedia, Hurwitz quaternion.
Index entries for "core" sequences.
Index entries for sequences related to Barnes-Wall lattices.
Index entries for sequences related to D_4 lattice.
Index entries for sequences related to Eisenstein series.

Cf. A002388, A000118, A000593, A008658, A096727, A108092, A108096.
Cf. A004016, A005882, A005928.
Partial sums give A046949.
Cf. A108092 (convolution fourth root).

Basis( ModularForms( Gamma0(2), 2), 54) [1]; /* Michael Somos, May 27 2014 */

readlib(ifactors): with(numtheory): for n from 1 to 100 do if n mod 2 = 0 then m := n/ifactors(n)[2][1][1]^ifactors(n)[2][1][2] else m := n fi: printf(`%d,`,24*sigma(m)) od: # James Sellers, Dec 07 2000

a[ n_] := If[ n < 0, 0, With[ {m = Floor @ Sqrt[4 n]}, SeriesCoefficient[ Sum[ q^( x^2 + y^2 + z^2 + t^2 + (x + y + z) t ), {x, -m, m}, {y, -m, m}, {z, -m, m}, {t, -m, m}] + O[q]^(n + 1), n]]]; (* Michael Somos, Jan 11 2011 *)
a[n_] := 24*Total[ Select[ Divisors[n], OddQ]]; a[0]=1; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Sep 12 2012 *)
a[ n_] := With[{m = InverseEllipticNomeQ @q}, SeriesCoefficient[ (1 + m) (EllipticK[ m] / (Pi/2))^2, {q, 0, n}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := With[{m = InverseEllipticNomeQ @q}, SeriesCoefficient[ (1 - m/2) (EllipticK[ m] / (Pi/2))^2, {q, 0, 2 n}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 4, 0, q]^4) / 2, {q, 0, 2 n}]; (* Michael Somos, Jun 04 2013 *)

{a(n) = if( n<1, n==0, 24 * sumdiv( n, d, d%2 * d))}; /* Michael Somos, Apr 17 2000 */

{a(n) = my(G); if( n<0, 0, G = [2, 1, 1, 1; 1, 2, 0, 0; 1, 0, 2, 0; 1, 0, 0, 2]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* Michael Somos, Sep 11 2007 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^20 / (eta(x + A) * eta(x^4 + A))^8 + 16 * x * eta(x^4 + A)^8 / eta(x^2 + A)^4, n))}; /* Michael Somos, Oct 21 2017 */

from sympy import divisors
def a(n): return 1 if n==0 else 24*sum(d for d in divisors(n) if d%2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017

from math import prod
from sympy import factorint
def A004011(n): return 24*prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if p > 2) if n else 1 # Chai Wah Wu, Nov 13 2022

ModularForms( Gamma0(2), 2, prec=54).0; # Michael Somos, Jun 04 2013

a(0) = 1; if n>0 then a(n) = 24 (Sum_{d|n, d odd, d>0} d) = 24 * A000593(n).
G.f.: 1 + 24 Sum_{n>0} n x^n /(1 + x^n). a(n) = A000118(2*n) = A096727(2*n).
G.f.: (1/2) * (theta_3(z)^4 + theta_4(z)^4) = theta_3(2z)^4 + theta_2(2z)^4 = Sum_{k>=0} a(k) * x^(2*k).
G.f.: Sum_{a, b, c, d in Z} x^(a^2 + b^2 + c^2 + d^2 + a*d + b*d + c*d). - Michael Somos, Jan 11 2011
Expansion of (1 + k^2) * K(k^2)^2 / (Pi/2)^2 in powers of nome q. - Michael Somos, Jun 10 2006
Expansion of (1 - k^2/2) * K(k^2)^2 / (Pi/2)^2 in powers of nome q^2. - Michael Somos, Mar 14 2012
Expansion of b(x) * b(x^2) + 3 * c(x) * c(x^2) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jan 11 2011
Expansion of b(x) * b(x^2) + c(x) * c(x^2) / 3 in powers of x^3 where b(), c() are cubic AGM theta functions. - Michael Somos, Mar 14 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 2 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 11 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 2*u*v - 7*v^2 - 8*v*w + 16*w^2. - Michael Somos, May 29 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 2*u1*u2 - 10*u1*u3 + 10*u1*u6 + 10*u2*u3 - 40*u2*u6 - 18*u3*u6. - Michael Somos, Sep 11 2007
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2 = 9.869604... (A002388). - Amiram Eldar, Dec 29 2023

Additional comments from Barry Brent (barryb(AT)primenet.com)

A261876 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (5x^2+7y^2+9z^2)y*z a square, where x,y,z,w are nonnegative integers with z > 0.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 01 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m (k = 0,1,2,... and m = 1, 7, 23, 647, 863).
(ii) For each triple (a,b,c) = (1,8,20), (3,5,15), (6,14,4), (7,29,5), (18,38,18), (39,81,51), (42,98,14), 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*(a*x^2+b*y^2+c*z^2) is a square.
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(4) = 1 since 4 = 0^2 + 0^2 + 2^2 + 0^2 with 2 > 0 and (5*0^2+7*0^2+9*2^2)*0*2 = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and (5*2^2+7*1^2+9*1^2)*1*1 = 6^2.
a(23) = 1 since 23 = 2^2 + 1^2 + 3^2 + 3^2 with 3 > 0 and (5*2^2+7*1^2+9*3^2)*1*3 = 18^2.
a(647) = 1 since 647 = 13^2 + 1^2 + 6^2 + 21^2 with 6 > 0 and (5*13^2+7*1^2+9*6^2)*1*6 = 84^2.
a(863) = 1 since 863 = 1^2 + 23^2 + 18^2 + 3^2 with 18 > 0 and (5*1^2+7*23^2+9*18^2)*23*18 = 1656^2.

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, A262357, A267121, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[y*z(5x^2+7y^2+9z^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,1,70}]

A267121 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with xyz(x+9y+11z+10w) a square, where x is a positive integer and y,z,w are nonnegative integers.

Original entry on oeis.org

1, 3, 2, 1, 6, 7, 1, 3, 7, 7, 6, 2, 6, 12, 1, 1, 12, 10, 7, 6, 13, 7, 2, 7, 8, 19, 8, 1, 18, 12, 2, 3, 14, 15, 13, 7, 7, 18, 1, 7, 25, 14, 6, 6, 19, 13, 2, 2, 14, 22, 12, 6, 18, 27, 4, 12, 13, 9, 19, 1, 18, 25, 5, 1, 24, 26, 6, 12, 26, 14, 2, 10, 16, 31, 16, 7, 24, 13, 4, 6

Offset: 1

Zhi-Wei Sun, May 01 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m (k = 0,1,2,... and m = 1, 7, 15, 39, 119, 127, 159, 239, 359, 391, 527, 543, 863, 5791).
(ii) Any positive integer can be written as x^2 + y^2 + z^2 + w^2 with 2*x*y*(x+2y+z+2w) (or 2*x*y*(x+6y+z+2w), or x*y*(x+11y+z+2w)) a square, where x,y,z,w are nonnegative integers with z > 0 (or w > 0).
(iii) 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*w+b*x+c*y+d*z) is a square, provided that (a,b,c,d) is among the following quadruples (1,1,2,3), (1,1,4,5), (1,1,6,9), (1,2,6,34), (1,3,6,m) (m = 12, 21, 27, 36), (1,3,9,18), (1,3,9,36), (1,3,18,27), (1,3,24,117), (1,3,90,99), (1,6,6,18), (1,6,6,30), (1,8,16,24).
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(4) = 1 since 4 = 2^2 + 0^2 + 0^2 + 0^2 with 2 > 0 and 2*0*0*(2+9*0+11*0+10*0) = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2 > 0 and 2*1*1*(2+9*1+11*1+10*1) = 8^2.
a(15) = 1 since 15 = 2^2 + 1^2 + 3^2 + 1^2 with 2 > 0 and 2*1*3*(2+9*1+11*3+10*1) = 18^2.
a(39) = 1 since 39 = 1^2 + 1^2 + 1^2 + 6^2 with 1 > 0 and 1*1*1*(1+9*1+11*1+10*6) = 9^2.
a(119) = 1 since 119 = 1^2 + 1^2 + 9^2 + 6^2 with 1 > 0 and 1*1*9*(1+9*1+11*9+10*6) = 39^2.
a(127) = 1 since 127 = 5^2 + 1^2 + 1^2 + 10^2 with 5 > 0 and 5*1*1*(5+9*1+11*1+10*10) = 25^2.
a(159) = 1 since 159 = 3^2 + 1^2 + 7^2 + 10^2 with 3 > 0 and 3*1*7*(3+9*1+11*7+10*10) = 63^2.
a(239) = 1 since 239 = 3^2 + 3^2 + 10^2 + 11^2 with 3 > 0 and 3*3*10*(3+9*3+11*10+10*11) = 150^2.
a(359) = 1 since 359 = 9^2 + 11^2 + 11^2 + 6^2 with 9 > 0 and 9*11*11*(9+9*11+11*11+10*6) = 561^2.
a(391) = 1 since 391 = 19^2 + 5^2 + 1^2 + 2^2 with 19 > 0 and 19*5*1*(19+9*5+11*1+10*2) = 95^2.
a(527) = 1 since 527 = 21^2 + 6^2 + 7^2 + 1^2 with 21 > 0 and 21*6*7*(21+9*6+11*7+10*1) = 378^2.
a(543) = 1 since 543 = 15^2 + 13^2 + 10^2 + 7^2 with 15 > 0 and 15*13*10*(15+9*13+11*10+10*7) = 780^2.
a(863) = 1 since 863 = 9^2 + 9^2 + 5^2 + 26^2 with 9 > 0 and 9*9*5*(9+9*9+11*5+10*26) = 405^2.
a(5791) = 1 since 5791 = 57^2 + 38^2 + 33^2 + 3^2 with 57 > 0 and 57*38*33*(57+9*38+11*33+10*3) = 7524^2.

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, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x*y*z(x+9y+11z+10*Sqrt[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}]

A000143 Number of ways of writing n as a sum of 8 squares.

Original entry on oeis.org

1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528, 56448, 74864, 78624, 84784, 109760, 143136, 154112, 149184, 194688, 261184, 252016, 246176, 327040, 390784, 390240, 395136, 476672, 599152, 596736, 550368, 693504, 859952

Offset: 0

N. J. A. Sloane

The relevant identity for the o.g.f. is theta_3(x)^8 = 1 + 16*Sum_{j >= 1} j^3*x^j/(1 - (-1)^j*x^j). See the Hardy-Wright reference, p. 315. - Wolfdieter Lang, Dec 08 2016

			1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); p. 79 Eq. (32.32).
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, pp. 314 - 315.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. M. Borwein and K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J. 7 (1-3) (2003) 95-127.
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.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See (5).
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.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

8th column of A286815. - Seiichi Manyama, May 27 2017
Row d=8 of A122141.
Cf. A008457, A035016, A010815.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A004018, A000118, A000141 for the expansion of the powers of 2, 4, 6 of theta_3(x).

# JacobiTheta3 is defined in A000122.
A000143List(len) = JacobiTheta3(len, 8)
A000143List(37) |> println # Peter Luschny, Mar 12 2018

(sum(x^(m^2),m=-10..10))^8;
with(numtheory); rJ := n-> if n=0 then 1 else 16*add((-1)^(n+d)*d^3, d in divisors(n)); fi; [seq(rJ(n),n=0..100)]; # N. J. A. Sloane, Sep 15 2018

Table[SquaresR[8, n], {n, 0, 33}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[8,Range[0,50]] (* Harvey P. Dale, Aug 26 2011 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]*QP[q^4])^2)^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^8, n))} /* Michael Somos, Sep 25 2005 */

from math import prod
from sympy import factorint
def A000143(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items())<<4 if n else 1 # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*8)
Q.representation_number_list(60) # Peter Luschny, Jun 20 2014

Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*Sum_{0

Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 21 2008

Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - Michael Somos, Sep 25 2005

G.f.: s(2)^40/(s(1)*s(4))^16, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]

Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos, Apr 10 2005

a(n) = 16 * b(n) and b(n) is multiplicative with b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) -2[p<3]. - Michael Somos, Sep 25 2005

G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - Michael Somos, Sep 25 2005

A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 16*(1 - 2^(1-s) + 4^(2-s))*zeta(s)*zeta(s-3). [Borwein and Choi], R. J. Mathar, Jul 02 2012

a(n) = (16/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 /24. - Vaclav Kotesovec, Jul 12 2024

A097057 Number of integer solutions to a^2 + b^2 + 2c^2 + 2d^2 = n.

Original entry on oeis.org

1, 4, 8, 16, 24, 24, 32, 32, 24, 52, 48, 48, 96, 56, 64, 96, 24, 72, 104, 80, 144, 128, 96, 96, 96, 124, 112, 160, 192, 120, 192, 128, 24, 192, 144, 192, 312, 152, 160, 224, 144, 168, 256, 176, 288, 312, 192, 192, 96, 228, 248, 288, 336, 216, 320, 288, 192, 320, 240, 240

Offset: 0

N. J. A. Sloane, Sep 15 2004

a^2 + b^2 + 2*c^2 + 2*d^2 is another (cf. A000118) of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos, Apr 01 2008

			1 + 4*q + 8*q^2 + 16*q^3 + 24*q^4 + 24*q^5 + 32*q^6 + 32*q^7 + 24*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 31.
Jesse Ira Deutsch, Bumby's technique and a result of Liouville on a quadratic form, Integers 8 (2008), no. 2, A2, 20 pp. MR2438287 (2009g:11047).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.29).
S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917), 11-21).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Jesse Ira Deutsch, A quaternionic proof of the representation formula of a quaternary quadratic form, J. Number Theory 113 (2005), no. 1, 149-174. MR2141762 (2006b:11033).
Y. Mimura, Almost Universal Quadratic Forms.
Olivia X. M. Yao and Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1-9. MR3141622.

a^2 + b^2 + 2*c^2 + m*d^2 = n: this sequence (m=2), A320124 (m=3), A320125 (m=4), A320126 (m=5), A320127 (m=6), A320128 (m=7), A320130 (m=8), A320131 (m=9), A320132 (m=10), A320133 (m=11), A320134 (m=12), A320135 (m=13), A320136 (m=14).

Cf. A000118, A004011, A008438, A097723, A112610.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}] (* Michael Somos, Jul 05 2011 *)
f[p_, e_] := (p^(e+1)-1)/(p-1); f[2, 1] = 2; f[2, e_] := 6; a[0] = 1; a[1] = 4; a[n_] := 4 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Aug 22 2023 *)

{a(n) = local(t); if( n<1, n>=0, t = 2^valuation( n, 2); 4 * sigma(n/t) * if( t>2, 6, t))} \\ Michael Somos, Sep 17 2004

{a(n) = local(A = x * O(x^n)); polcoeff( (eta(x^2 + A) * eta(x^4 + A))^6 / (eta(x + A) * eta(x^8 + A))^4, n)} \\ Michael Somos, Sep 17 2004

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0, 0; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], n)[n])} \\ Michael Somos, Oct 29 2005

A097057(n)=if(n,sigma(n>>n=valuation(n,2))*if(n>1,24,4<M. F. Hasler, May 07 2018

Euler transform of period 8 sequence [4, -2, 4, -8, 4, -2, 4, -4, ...]. - Michael Somos, Sep 17 2004
Multiplicative with a(n) = 4*b(n), b(2) = 2, b(2^e) = 6 if e > 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2. - Michael Somos, Sep 17 2004
Expansion of (eta(q^2) * eta(q^4))^6 / (eta(q) * eta(q^8))^4 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 05 2011
G.f.: (theta_3(q) * theta_3(q^2))^2.
G.f.: Product_{k>0} ((1-x^(2k))(1-x^(4k)))^6/((1-x^k)(1-x^(8k)))^4.
G.f.: 1 + Sum_{k>0} 8 * x^(4*k) / (1 + x^(4*k))^2 + 4 * x^(2*k-1) / (1 - x^(2*k-1))^2 = 1 + Sum_{k>0} (2 + (-1)^k) * 4*k * x^(2*k) / (1 + x^(2*k)) + 4*(2*k - 1) * x^(2*k-1) / (1 - x^(2*k - 1)). - Michael Somos, Oct 22 2005
a(2*n) = A000118(n). a(2*n + 1) = 4 * A008438(n). a(4*n) = A004011(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 2) = 8 * A008438(n). a(4*n + 3) = 16 * A097723(n). - Michael Somos, Jul 05 2011

Added keyword mult and minor edits by M. F. Hasler, May 07 2018

A268197 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w(25w + 24x + 48y + 96*z) a square, where w is a positive integer and x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 04 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 7, 15, 23, 43, 55, 463, 4^k*m (k = 0,1,2,... and m = 1, 31, 34).
(ii) For each triple (a,b,c) = (1,3,4), (2,3,4), (2,4,6), any positive integer can be written as w^2 + x^2 + y^2 + z^2 with w*(25*w + 24*(a*x+b*y+c*z)) a square, where w is a positive integer and x,y,z are nonnegative integers.
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 > 0 and 1*(25*1 + 24*0 + 48*0 + 96*0) = 5^2.
a(2) = 2 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 1*(25*1 + 24*0 + 48*0 + 96*1) = 11^2, and also 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*0 + 96*0) = 7^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 1 > 0 and 1*(25*1 + 24*0 + 48*1 + 96*1) = 13^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*1 + 96*2) = 17^2.
a(15) = 1 since 15 = 1^2 + 3^2 + 2^2 + 1^2 with 1 > 0 and 1*(25*1 + 24*3 + 48*2 + 96*1) = 17^2.
a(23) = 1 since 23 = 3^2 + 2^2 + 3^2 + 1^2 with 3 > 0 and 3*(25*3 + 24*2 + 48*3 + 96*1) = 33^2.
a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*2 + 96*5) = 25^2.
a(34) = 1 since 34 = 1^2 + 1^2 + 4^2 + 4^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*4 + 96*4) = 25^2.
a(43) = 1 since 43 = 3^2 + 3^2 + 3^2 + 4^2 with 3 > 0 and 3*(25*3 + 24*3 + 48*3 + 96*4) = 45^2.
a(55) = 1 since 55 = 3^2 + 1^2 + 6^2 + 3^2 with 3 > 0 and 3*(25*3 + 24*1 + 48*6 + 96*3) = 45^2.
a(463) = 1 since 463 = 3^2 + 18^2 + 11^2 + 3^2 with 3 > 0 and 3*(25*3 + 24*18 + 48*11 + 96*3) = 63^2.

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.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[25x^2+24x(y+2z+4*Sqrt[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,1,70}]

A270073 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with xy + 2z*w a square, where x,y,z are nonnegative integers and w is an integer with x <= y and z >= |w|.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, May 07 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 3, 11, 15, 23, 43, 67, 79, 155, 211, 331, 347, 403, 427, 659, 899, 1443, 1955, 2^k*m (k = 0,1,2,... and m = 14, 35, 62, 158, 382).
(ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x*y + z*w/2 a square, where x,y,z are nonnegative integers and w is an integer with 2 | z*w and x <= y >= |w| <= z.
We have verified that a(n) > 0 for all n = 0,...,10^5.
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(3) = 1 since 3 = 1^2 + 1^2 + 1^2 + 0^2 with 1 = 1, 1 > 0 and 1*1 + 2*1*0 = 1^2.
a(11) = 1 since 11 = 1^2 + 1^2 + 3^2 + 0^2 with 1 = 1, 3 > 0 and 1*1 + 2*3*0 = 1^2.
a(14) = 1 since 14 = 0^2 + 3^2 + 2^2 + 1^2 with 0 < 3, 2 > 1 and 0*3 + 2*2*1 = 2^2.
a(15) = 1 since 15 = 2^2 + 3^2 + 1^2 + (-1)^2 with 2 < 3, 1 = |-1| and 2*3 + 2*1*(-1) = 2^2.
a(23) = 1 since 23 = 2^2 + 3^2 + 3^2 + (-1)^2 with 2 < 3, 3 > |-1| and 2*3 + 2*3*(-1) = 0^2.
a(35) = 1 since 35 = 3^2 + 3^2 + 4^2 + (-1)^2 with 3 = 3, 4 > |-1| and 3*3 + 2*4*(-1) = 1^2.
a(43) = 1 since 43 = 3^2 + 3^2 + 5^2 + 0^2 with 3 = 3, 5 > 0 and 3*3 + 2*5*0 = 3^2.
a(62) = 1 since 62 = 3^2 + 4^2 + 6^2 + (-1)^2 with 3 < 4, 6 > |-1| and 3*4 + 2*6*(-1) = 0^2.
a(67) = 1 since 67 = 3^2 + 3^2 + 7^2 + 0^2 with 3 = 3, 7 > 0 and 3*3 + 2*7*0 = 3^2.
a(79) = 1 since 79 = 2^2 + 7^2 + 5^2 + (-1)^2 with 2 < 7, 5 > |-1| and 2*7 + 2*5*(-1) = 2^2.
a(155) = 1 since 155 = 3^2 + 11^2 + 4^2 + (-3)^2 with 3 < 11, 4 > |-3| and 3*11 + 2*4*(-3) = 3^2.
a(158) = 1 since 158 = 1^2 + 12^2 + 3^2 + (-2)^2 with 1 < 12, 3 > |-2| and 1*12 + 2*3*(-2) = 0^2.
a(211) = 1 since 211 = 9^2 + 9^2 + 7^2 + 0^2 with 9 = 9, 7 > 0 and 9*9 + 2*7*0 = 9^2.
a(331) = 1 since 331 = 9^2 + 9^2 + 13^2 + 0^2 with 9 = 9, 13 > 0 and 9*9 + 2*13*0 = 9^2.
a(347) = 1 since 347 = 13^2 + 13^2 + 3^2 + 0^2 with 13 = 13, 3 > 0 and 13*13 + 2*3*0 = 13^2.
a(382) = 1 since 382 = 5^2 + 16^2 + 10^2 + 1^2 with 5 < 16, 10 > 1 and 5*16 + 2*10*1 = 10^2.
a(403) = 1 since 403 = 13^2 + 13^2 + 7^2 + 4^2 with 13 = 13, 7 > 4 and 13*13 + 2*7*4 = 15^2.
a(427) = 1 since 427 = 11^2 + 11^2 + 13^2 + 4^2 with 11 = 11, 13 > 4 and 11*11 + 2*13*4 = 15^2.
a(659) = 1 since 659 = 17^2 + 17^2 + 9^2 + 0^2 with 17 = 17, 9 > 0 and 17*17 + 2*9*0 = 17^2.
a(899) = 1 since 899 = 7^2 + 15^2 + 24^2 + 7^2 with 7 < 15, 24 > 7 and 7*15 + 2*24*7 = 21^2.
a(1443) = 1 since 1443 = 7^2 + 31^2 + 17^2 + 12^2 with 7 < 31, 17 > 12 and 7*31 + 2*17*12 = 25^2.
a(1955) = 1 since 1955 = 19^2 + 27^2 + 28^2 + (-9)^2 with 19 < 27, 28 > |-9| and 19*27 + 2*28*(-9) = 3^2.

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.
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, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620.

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

All statements in the examples checked by Rick L. Shepherd, May 27 2016

Previous Showing 41-50 of 199 results. Next

cp's OEIS Frontend

A301391 Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y even such that x^2 - (6*y)^2 = 4^k for some k = 0,1,2,....

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A260625 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+3*y+13*z)*x*y*z a square, where x is a positive integer, and y,z,w are nonnegative integers with y >= z.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A272620 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w + x + y - z a square, where w is an integer and x,y,z are nonnegative integers with |w| <= x >= y <= z < x + y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Sage

Formula

Extensions

A261876 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (5*x^2+7*y^2+9*z^2)*y*z a square, where x,y,z,w are nonnegative integers with z > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A267121 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x*y*z*(x+9*y+11*z+10*w) a square, where x is a positive integer and y,z,w are nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A000143 Number of ways of writing n as a sum of 8 squares.

Views

Author

Keywords

Comments

Examples

References

A260625 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+3y+13z)xy*z a square, where x is a positive integer, and y,z,w are nonnegative integers with y >= z.

A261876 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (5x^2+7y^2+9z^2)y*z a square, where x,y,z,w are nonnegative integers with z > 0.

A267121 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with xyz(x+9y+11z+10w) a square, where x is a positive integer and y,z,w are nonnegative integers.

A097057 Number of integer solutions to a^2 + b^2 + 2c^2 + 2d^2 = n.

A268197 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w(25w + 24x + 48y + 96*z) a square, where w is a positive integer and x,y,z are nonnegative integers.

A270073 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with xy + 2z*w a square, where x,y,z are nonnegative integers and w is an integer with x <= y and z >= |w|.