A236931 -id:A236931 - cp's OEIS frontend

A000118 Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.

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

N. J. A. Sloane

a^2 + b^2 + c^2 + d^2 is one of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos, Apr 01 2008
a(n) is also the number of quaternions q = a + bi + cj + dk, where a, b, c, d are integers, such that a^2 + b^2 + c^2 + d^2 = n (i.e., so that n is the norm of q). These are Lipschitz integer quaternions. - Rick L. Shepherd, Mar 27 2009
Number 5 and 35 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018
This is the convolution square of A004018. - Pierre Abbat, May 15 2023

			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 + ...
a(1)=8 counts 1 = 1^2 + 0^2 + 0^2 + 0^2 = 0^2 + 1^2 + 0^2 + 0^2 = 0^2 + 0^2 + 1^2 + 0^2 = 0^2 + 0^2 + 0^2 + 1^2 and 4 more sums where 1^2 is replaced by (-1)^2. - _R. J. Mathar_, May 16 2023

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, ch. 8, pp. 231-2.
J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, Springer-Verlag, p. 108, Eq. (49).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.28). See also top of p. 94.
E. Freitag and R. Busam, Funktionentheorie 1, 4. Auflage, Springer, 2006, p. 392.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314, Theorem 386.
Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of integers, Chapman & Hall/CRC, 2006, p. 29.
S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917) 11-21).

N. J. A. Sloane, Table of n, a(n) for n = 0..50000 (first 10000 terms from T. D. Noe)
George E. Andrews, S. B. Ekhad, and D. Zeilberger A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares, arXiv:math/9206203 [math.CO], 1992.
George E. Andrews, S. B. Ekhad, and D. Zeilberger, A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a sum of Four Squares
George E. Andrews, Sumit Kumar Jha, and J. López-Bonilla, Sums of Squares, Triangular Numbers, and Divisor Sums, Journal of Integer Sequences, Vol. 26 (2023), Article 23.2.5.
Michael Ball and Dario Alejandro Alpern, Every positive integer is a sum of four integer squares
Cristina Ballantine and Mircea Merca, Jacobi's four and eight squares theorems and partitions into distinct parts, Mediterr. J. Math 16 (2009) 26.
R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
R. T. Bumby, Sums of four squares [Cached copy]
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.
Peter L. Clark, A theorem of Minkowski; the four squares theorem (no date).
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
E. Grosswald, Representations of Integers as Sums of an Even Number of Squares, Springer-Verlag, NY, 1985, p. 121.
M. D. Hirschhorn, A Simple Proof of Jacobi's Four-Square Theorem, Proceedings of the American Mathematical Society, Vol. 101, No. 3 (Nov., 1987), pp. 436-438
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.]
G. Nebe and N. J. A. Sloane, The lattice Z4
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.
Y. Mimura, Almost Universal Quadratic Forms.
Simon Plouffe, Table of n, a(n) for n=0..105817
B. K. Spearman and K. S. Williams, The simplest arithmetic proof of Jacobi's four squares theorem, Far East Journal of Mathematical Sciences 2.3 (2000): 433-440.
Eric van Fossen Conrad, Jacobi's Four Square Theorem
Min Wang and Zhi-Hong Sun, On the number of representations of n as a linear combination of four triangular numbers II, arXiv:1511.00478 [math.NT], 2015.
Eric W. Weisstein, Quaternion Norm.
Wikipedia, Hurwitz quaternion
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.
Index entries for sequences related to sums of squares

Row d=4 of A122141 and of A319574, 4th column of A286815.
Cf. A000122, A000203, A000265, A004011, A005879, A046897, A096727.
For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1, 2, 3, 4, 5, 6, 7, 8, 12, see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

a000118 0 = 1
a000118 n = 8 * a046897 n  -- Reinhard Zumkeller, Aug 12 2015

# JacobiTheta3 is defined in A000122.
A000118List(len) = JacobiTheta3(len, 4)
A000118List(57) |> println # Peter Luschny, Mar 12 2018

a(n) = 8 * sum(find(mod(n,1:n)==0 & mod(1:n,4))) + (n==0) % David Mellinger, Aug 04 2025

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

(add(q^(m^2),m=-10..10))^4; seq(coeff(%,q,n), n=0..50);
# Alternative:
A000118list := proc(len) series(JacobiTheta3(0, x)^4, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000118list(57); # Peter Luschny, Oct 02 2018

Table[SquaresR[4, n], {n, 0, 46}]
a[ n_] :=  SeriesCoefficient[ EllipticTheta[ 3, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 8 Sum[ If[ Mod[ d, 4] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, Feb 20 2015 *)
QP = QPochhammer; CoefficientList[QP[-q]^8/QP[q^2]^4 + O[q]^60, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<1, n==0, 8 * sumdiv( n, d, if( d%4, d)))}; /* Michael Somos, Apr 01 2003 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^2))^4, n))}; /* Michael Somos, Apr 01 2008 */

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

a(n) = 8*sigma(n) - if (n % 4, 0, 32*sigma(n/4)); \\ Michel Marcus, Jul 13 2016

from sympy import divisors
def a(n): return 1 if n==0 else 8*sum(d for d in divisors(n) if d%4 != 0)
print([a(n) for n in range(57)]) # Michael S. Branicky, Jan 08 2021

from sympy import divisor_sigma
def A000118(n): return 1 if n == 0 else 8*divisor_sigma(n) if n % 2 else 24*divisor_sigma(int(bin(n)[2:].rstrip('0'),2)) # Chai Wah Wu, Jun 27 2022

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

Q = DiagonalQuadraticForm(ZZ, [1]*4)
Q.representation_number_list(60) # Peter Luschny, Jun 20 2014

G.f.: theta_3(q)^4 = (Product_{n>=1} (1-q^(2n))*(1+q^(2n-1))^2)^4 = eta(-q)^8/eta(q^2)^4; eta = Dedekind's function.
a(n) = 8*sigma(n) - 32*sigma(n/4) for n > 0, where the latter term is 0 if n is not a multiple of 4.
Euler transform of period 4 sequence [8, -12, 8, -4, ...]. - Michael Somos, Dec 16 2002
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). - Michael Somos, Nov 02 2006
G.f. is a period 1 Fourier series which satisfies f(-1/(4*t)) = 4*(t/i)^2*f(t) where q = exp(2*Pi*i*t). - Michael Somos, Jan 25 2008
For n > 0, a(n)/8 is multiplicative and a(p^n)/8 = 1 + p + p^2 + ... + p^n for p an odd prime, a(2^n)/8 = 1 + 2 for n > 0.
a(n) = 8*A000203(n/A006519(n))*(2 + (-1)^n). - Benoit Cloitre, May 16 2002
G.f.: 1 + 8*Sum_{k>0} x^k / (1 + (-x)^k)^2 = 1 + 8*Sum_{k>0} k * x^k / (1 + (-x)^k).
G.f. = s(2)^20/(s(1)*s(4))^8, where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
Fine gives another explicit formula for a(n) in terms of the divisors of n.
a(n) = 8*A046897(n), n > 0. - Ralf Stephan, Apr 02 2003
A096727(n) = (-1)^n * a(n). a(2*n) = A004011(n). a(2*n + 1) = A005879(n).
Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 8*(1-4^(1-s))*zeta(s)*zeta(s-1). [Ramanu. J. 7 (2003) 95-127, eq (3.2)]. - R. J. Mathar, Jul 02 2012
Average value is (Pi^2/2)*n + O(sqrt(n)). - Charles R Greathouse IV, Feb 17 2015
From Wolfdieter Lang, Jan 14 2016: (Start)
For n >= 1: a(n) = 8*Sum_{d | n} b(d)*d, with b(d) = 1 if d/4 is not an integer else 0. See, e.g., the Freitag-Busam reference, p. 392.
For n >= 1: a(n) = 8*sigma(n) if n is odd else 24*sigma(m(n)), where m(n) is the largest odd divisor of n (see A000265), and sigma is given in A000203. See the Moreno-Wagstaff reference, Theorem 2. 6 (Jacobi), p. 29. (End)
a(n) = (8/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

A236923 Number of integer solutions to a^2 + b^2 + c^2 + 4*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 8, 36, 48, 16, 24, 78, 72, 24, 32, 84, 96, 48, 24, 108, 156, 40, 48, 192, 144, 48, 96, 186, 168, 80, 64, 180, 288, 64, 24, 288, 216, 96, 104, 228, 240, 112, 144, 252, 384, 88, 96, 468, 288, 96, 96, 342, 372, 144, 112, 324, 480, 144, 192, 480, 360, 120, 192, 372, 384, 208, 24, 504, 576, 136, 144, 576, 576, 144, 312, 444, 456, 248, 160, 576, 672

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 and Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622.

Cf. A097057, A236924.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

with(numtheory);
s:=n-> if whattype(n) = integer then sigma(n) else 0; fi;
f:=proc(n) global s;
  if (n mod 4) = 0 then 8*s(n/4)-32*s(n/16)
elif (n mod 4) = 2 then 12*s(n/2)
elif (n mod 4) = 3 then 2*s(n)
else 6*s(n);
fi; end;
[seq(f(n),n=1..100)];
# a(0)=1 must be added separately

EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, q^4] + O[q]^80 // CoefficientList[#, q]& (* Jean-François Alcover, Mar 04 2023, after Ilya Gutkovskiy *)

See Maple code.

G.f.: theta_3(q)^3*theta_3(q^4), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A236928 Number of integer solutions to a^2 + b^2 + c^2 + 2*d^2 = n.

Original entry on oeis.org

1, 6, 14, 20, 30, 40, 36, 48, 62, 42, 72, 100, 68, 120, 112, 48, 126, 108, 98, 180, 136, 160, 180, 144, 132, 126, 216, 200, 240, 280, 112, 192, 254, 120, 252, 320, 210, 360, 324, 144, 264, 252, 288, 420, 340, 280, 336, 288, 260, 342, 294, 360, 408, 520, 360, 240, 496

Offset: 0

N. J. A. Sloane, Feb 15 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000
I. J. Zucker, Exact Evaluation of Some New Lattice Sums, Symmetry, 2017, 9(12), 314.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1, 2, 3, 4, 5, 6, 7, 8, 12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^2), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

G.f.: 1 + 8*Sum{n >= 1} n*(q^n - q^(3*n) - q^(5*n) + q^(7*n))/(1 - q^(8*n)) - 2*Sum_{n >= 0} (-1)^((n^2+n)/2)*(2*n+1)q^(2*n+1)/(1 - q^(2*n+1)). See Zucker p. 5. Cf. A117000. - Peter Bala, Feb 25 2021

A236926 Number of integer solutions to a^2 + b^2 + c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 6, 12, 10, 18, 48, 40, 12, 60, 78, 24, 48, 70, 84, 120, 32, 66, 192, 84, 36, 144, 180, 120, 96, 136, 126, 168, 82, 84, 336, 200, 60, 252, 288, 96, 96, 234, 228, 360, 140, 120, 480, 144, 84, 336, 336, 240, 192, 310, 258, 252, 128, 252, 624, 400, 96, 408, 540, 168

Offset: 0

N. J. A. Sloane, Feb 15 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
J. Liouville, Sur La Forme x^2+y^2+z^2+3t^2, Journal de Mathématiques Pures et appliquées (2) 8, 193-204 (1863).

Cf. A236924.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^3), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A236930 Number of integer solutions to a^2 + b^2 + c^2 + 5*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 26, 36, 24, 28, 42, 72, 72, 8, 48, 108, 48, 54, 64, 84, 120, 26, 72, 144, 88, 84, 126, 216, 80, 24, 180, 156, 192, 92, 96, 288, 144, 42, 144, 240, 144, 168, 252, 144, 168, 72, 182, 396, 184, 72, 258, 372, 192, 48, 208, 360, 312, 252, 160, 360, 360, 48

Offset: 0

N. J. A. Sloane, Feb 16 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

Maple
```
See A236924.
```

Join[{1},Table[Length[Reduce[a^2+b^2+c^2+5d^2==n,{a,b,c,d},Integers]],{n,60}]] (* Harvey P. Dale, Jul 02 2017 *)

G.f.: theta_3(q)^3*theta_3(q^5), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A236932 Number of integer solutions to a^2 + b^2 + c^2 + 7*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 2, 24, 54, 40, 36, 56, 72, 48, 24, 66, 96, 84, 40, 72, 144, 24, 12, 120, 102, 120, 80, 98, 132, 72, 64, 84, 240, 160, 48, 198, 180, 120, 72, 136, 240, 240, 84, 84, 312, 120, 96, 248, 246, 180, 96, 216, 228, 240, 80, 156, 360, 216, 120, 168

Offset: 0

N. J. A. Sloane, Feb 16 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

For number of solutions to a^2 + b^2 + c^2 + k*d^2 = n for k = 1, 2, 3, 4, 5, 6, 7, 8, 12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^7), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A236933 Number of integer solutions to a^2 + b^2 + c^2 + 12*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 10, 36, 72, 16, 18, 96, 84, 24, 48, 108, 72, 48, 40, 78, 168, 32, 12, 168, 120, 48, 60, 144, 96, 48, 78, 84, 216, 64, 24, 240, 144, 24, 48, 168, 144, 96, 70, 114, 252, 64, 84, 312, 240, 48, 120, 252, 168, 120, 32

Offset: 0

N. J. A. Sloane, Feb 16 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Different from A005875.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^12), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 03 2018

A236927 Number of integer solutions to a^2 + b^2 + c^2 + 8*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 0, 14, 42, 48, 40, 20, 72, 96, 0, 30, 108, 84, 72, 40, 96, 120, 0, 36, 126, 144, 80, 48, 168, 96, 0, 62, 120, 216, 128, 42, 216, 216, 0, 72, 252, 192, 168, 100, 168, 288, 0, 68, 342, 252, 144, 120, 312, 240, 0, 112, 216, 336, 232, 48, 360, 384

Offset: 0

N. J. A. Sloane, Feb 15 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Liouville, J., Sur la forme X^2 + Y^2 + Z^2 + 8T^2. Journal de mathématiques pures et appliquées 2e série, tome 6 (1861), p. 324-328.

Cf. A236924.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^8), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A319822 Number of solutions to x^2 + 2y^2 + 5z^2 + 5*w^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 12, 8, 18, 14, 4, 28, 12, 24, 32, 0, 34, 20, 14, 28, 4, 32, 44, 40, 28, 10, 40, 56, 64, 72, 8, 48, 66, 24, 68, 8, 46, 88, 60, 32, 4, 52, 64, 116, 76, 12, 64, 72, 60, 82, 26, 72, 104, 104, 88, 8, 112, 56, 136, 140, 8, 136, 96, 72, 98, 16, 72, 132

Offset: 0

Jianing Song, Sep 28 2018

nonn

Ramanujan (1917) claimed that there are exactly 55 possible choice for a <= b <= c <= d such that a*x^2 + b*y^2 + c*z^2 + d*w^2 represents all natural numbers, but L. E. Dickson (1927) has pointed out that Ramanujan has overlooked the fact that (1, 2, 5, 5) does not represent 15. Consequently, there are only 54 forms. This sequence is related to the form (1, 2, 5, 5). As is proven, a(n) = 0 iff n = 15.

There are also many (a, b, c, d) other than this such that a*x^2 + b*y^2 + c*z^2 + d*w^2 represents all but finitely many natural numbers. For example, x^2 + y^2 + 5*z^2 + 5*w^2 represents all natural numbers except for 3 (cf. A236929); x^2 + y^2 + z^2 + d*w^2 (d == 2 (mod 4) or d = 9, 17, 25, 36, 68, 100 and some others) represents all natural numbers except for those of the form 4^i*(8*j + 7) and < d; x^2 + 2*y^2 + 6*z^2 + d*w^2 (d == 2 (mod 4) or d = 11, 19 and some others) represents all natural numbers except for those of the form 4^i*(8*j + 5) and < d.

			a(5) = 4 because 0^2 + 2*0^2 + 5*0^2 + 5*1^2 = 0^2 + 2*0^2 + 5*0^2 + 5*(-1)^2 = 0^2 + 2*0^2 + 5*1^2 + 5*0^2 = 0^2 + 2*0^2 + 5*(-1)^2 + 5*0^2 = 5 and these are the only four solutions to x^2 + 2*y^2 + 5*z^2 + 5*w^2 = 5.

J. H. Conway, Universal quadratic forms and the fifteen theorem, Contemporary Mathematics 272 (1999), 23-26.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bulletin of the American Mathematical Society, 1927, 33(1):63-70.
H. D. Kloosterman, On the representation of numbers in the form ax^2 + by^2 + cz^2 + dt^2, Acta Mathematica, 1927, 49(3-4):407-464.
S. Ramanujan, On the expression of a number in the form ax^2 + by^2 + cz^2 + du^2, Proc. Camb. Phil. Soc. 19 (1917), 11-21.

Cf. A004018, A033715, A236922-A236933.
From Seiichi Manyama, Oct 07 2018: (Start)
possible choice:
  k   |   a, b, c,  d   | Number of solutions
------+-----------------+--------------------
|   1, 1, 1,  1   | A000118
|   1, 1, 1,  2   | A236928
|   1, 1, 1,  3   | A236926
|   1, 1, 1,  4   | A236923
|   1, 1, 1,  5   | A236930
|   1, 1, 1,  6   | A236931
|   1, 1, 1,  7   | A236932
|   1, 1, 2,  2   | A097057
|   1, 1, 2,  3   | A320124
|   1, 1, 2,  4   | A320125
|   1, 1, 2,  5   | A320126
|   1, 1, 2,  6   | A320127
|   1, 1, 2,  7   | A320128
|   1, 1, 2,  8   | A320130
|   1, 1, 2,  9   | A320131
|   1, 1, 2, 10   | A320132
|   1, 1, 2, 11   | A320133
|   1, 1, 2, 12   | A320134
|   1, 1, 2, 13   | A320135
|   1, 1, 2, 14   | A320136
|   1, 1, 3,  3   | A034896
|   1, 1, 3,  4   | A272364
|   1, 1, 3,  5   | A320147
|   1, 1, 3,  6   | A320148
|   1, 2, 2,  2   | A320149
|   1, 2, 2,  3   | A320150
|   1, 2, 2,  4   | A236924
|   1, 2, 2,  5   | A320151
|   1, 2, 2,  6   | A320152
|   1, 2, 2,  7   | A320153
|   1, 2, 3,  3   | A320138
|   1, 2, 3,  4   | A320139
|   1, 2, 3,  5   | A320140
|   1, 2, 3,  6   | A033712
|   1, 2, 3,  7   | A320188
|   1, 2, 3,  8   | A320189
|   1, 2, 3,  9   | A320190
|   1, 2, 3, 10   | A320191
|   1, 2, 4,  4   | A320193
|   1, 2, 4,  5   | A320194
|   1, 2, 4,  6   | A320195
|   1, 2, 4,  7   | A320196
|   1, 2, 4,  8   | A033720
|   1, 2, 4,  9   | A320197
|   1, 2, 4, 10   | A320198
|   1, 2, 4, 11   | A320199
|   1, 2, 4, 12   | A320200
|   1, 2, 4, 13   | A320201
|   1, 2, 4, 14   | A320202
|   1, 2, 5,  6   | A320163
|   1, 2, 5,  7   | A320164
|   1, 2, 5,  8   | A320165
|   1, 2, 5,  9   | A320166
|   1, 2, 5, 10   | A033722
(End)

JT := (k, n) -> JacobiTheta3(0, x^k)^n:
A319822List := proc(len) series(JT(1,1)*JT(2,1)*JT(5,2), x, len+1);
seq(coeff(%, x, j), j=0..len) end: A319822List(67); # Peter Luschny, Oct 01 2018

CoefficientList[EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^2] EllipticTheta[ 3, 0, q^5]^2 + O[q]^100, q] (* Jean-François Alcover, Jun 15 2019 *)

A004018(n) = if(n, 4*sumdiv(n,d,kronecker(-4,d)), 1);
A033715(n) = if(n, 2*sumdiv(n,d,kronecker(-2,d)), 1);
a(n) = my(i=0); for(k=0, n\5, i+=A004018(k)*A033715(n-5*k)); i

N=99; q='q+O('q^N);
gf = (eta(q^2)*eta(q^4))^3*eta(q^10)^10/(eta(q)*eta(q^5)^2*eta(q^8)*eta(q^20)^2)^2;
Vec(gf) \\ Altug Alkan, Oct 01 2018

Q = DiagonalQuadraticForm(ZZ, [1, 2, 5, 5])
Q.theta_series(68).list() # Peter Luschny, Oct 01 2018

a(n) = Sum_{k=0..floor(n/5)} A004018(k)*A033715(n-5*k).

G.f.: theta_3(q)*theta_3(q^2)*theta_3(q^5)^2, where theta_3() is the Jacobi theta function.

cp's OEIS Frontend

A000118 Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.

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A236923 Number of integer solutions to a^2 + b^2 + c^2 + 4*d^2 = n.

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A236928 Number of integer solutions to a^2 + b^2 + c^2 + 2*d^2 = n.

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A236926 Number of integer solutions to a^2 + b^2 + c^2 + 3*d^2 = n.

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A236930 Number of integer solutions to a^2 + b^2 + c^2 + 5*d^2 = n.

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A236932 Number of integer solutions to a^2 + b^2 + c^2 + 7*d^2 = n.

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A236933 Number of integer solutions to a^2 + b^2 + c^2 + 12*d^2 = n.

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A319822 Number of solutions to x^2 + 2y^2 + 5z^2 + 5*w^2 = n.