A033712 -id:A033712 - cp's OEIS frontend

A320139 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 4*d^2 = n.

1, 2, 2, 6, 8, 8, 16, 16, 14, 22, 24, 16, 22, 32, 12, 32, 44, 16, 42, 52, 36, 40, 64, 32, 40, 86, 24, 50, 72, 40, 60, 80, 38, 48, 112, 48, 72, 96, 64, 80, 120, 64, 48, 124, 52, 104, 96, 64, 106, 110, 110, 96, 144, 72, 128, 160, 60, 132, 120, 64, 144, 160, 60, 112, 164

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

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

Cf. A320138, A320140, A033712, A320188, A320189, A320190, A320191.

CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 4}], {q, 0, 70}], q] (* G. C. Greubel, Oct 29 2018 *)

q='q+O('q^70); Vec(prod(k=1,4, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )) \\ G. C. Greubel, Oct 29 2018

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^4).

A320138 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 2, 2, 8, 10, 8, 24, 16, 10, 38, 8, 12, 48, 8, 32, 64, 26, 36, 70, 28, 24, 80, 28, 48, 96, 42, 40, 76, 48, 24, 112, 64, 58, 160, 68, 32, 126, 56, 44, 192, 56, 84, 176, 44, 60, 88, 80, 96, 208, 114, 74, 176, 72, 72, 172, 80, 112, 288, 88, 76, 224, 72, 112, 304, 90, 96

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

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

Cf. A320139, A320140, A033712, A320188, A320189, A320190, A320191.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3)^2.

A320140 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 5*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 6, 16, 8, 14, 26, 8, 32, 26, 8, 40, 16, 22, 40, 22, 32, 46, 40, 24, 48, 40, 42, 72, 50, 32, 64, 56, 28, 74, 48, 60, 112, 78, 24, 72, 76, 40, 144, 48, 48, 120, 50, 52, 48, 70, 98, 150, 128, 40, 84, 128, 52, 176, 120, 56, 208, 96, 72, 92, 72, 102, 192, 156

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

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

Cf. A320138, A320139, A033712, A320188, A320189, A320190, A320191.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^5).

A320188 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 7*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 12, 6, 6, 18, 12, 20, 26, 28, 20, 20, 34, 4, 30, 44, 20, 48, 44, 20, 20, 38, 16, 42, 62, 40, 56, 48, 34, 24, 72, 40, 70, 112, 36, 56, 68, 44, 40, 124, 60, 60, 124, 24, 66, 62, 54, 96, 92, 80, 64, 80, 64, 88, 136, 64, 76, 140, 52, 70, 166, 44, 104, 196, 44

Offset: 0

Seiichi Manyama, Oct 07 2018

nonn

a(n) > 0 for n >= 0.

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

Cf. A320138, A320139, A320140, A033712, A320189, A320190, A320191.

CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^7], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(14))^5/(eta(q^7)* eta(q^(28)))^2 ) \\ G. C. Greubel, Oct 29 2018

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^7).

A320189 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 8*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 12, 4, 4, 18, 4, 20, 30, 12, 36, 24, 10, 32, 14, 24, 48, 32, 36, 40, 24, 18, 28, 34, 36, 60, 60, 28, 28, 40, 16, 56, 78, 44, 108, 68, 8, 72, 24, 40, 144, 60, 72, 112, 30, 46, 42, 64, 84, 116, 120, 40, 72, 84, 28, 116, 96, 60, 180, 68, 34, 120, 60, 64, 192, 80

Offset: 0

Seiichi Manyama, Oct 07 2018

nonn

a(n) > 0 for n >= 0.

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

Cf. A320138, A320139, A320140, A033712, A320188, A320190, A320191.

CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^8], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(16))^5/(eta(q^8)* eta(q^(32)))^2 ) \\ G. C. Greubel, Oct 29 2018

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^8).

A320190 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 9*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 12, 4, 2, 16, 4, 12, 30, 16, 20, 40, 14, 8, 42, 8, 28, 60, 20, 32, 44, 22, 8, 46, 28, 36, 84, 36, 18, 40, 32, 8, 80, 40, 40, 120, 36, 24, 56, 40, 44, 140, 48, 48, 126, 22, 62, 64, 40, 68, 132, 72, 52, 120, 52, 60, 136, 56, 36, 160, 46, 64, 180, 40, 48

Offset: 0

Seiichi Manyama, Oct 07 2018

nonn

a(n) > 0 for n >= 0.

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

Cf. A320138, A320139, A320140, A033712, A320188, A320189, A320191.

CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^9], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(18))^5/(eta(q^9)* eta(q^(36)))^2 ) \\ G. C. Greubel, Oct 29 2018

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^9).

A320191 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 10*d^2 = n.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 12, 4, 2, 14, 2, 12, 22, 16, 24, 24, 30, 12, 18, 36, 12, 40, 48, 16, 36, 42, 12, 26, 40, 28, 60, 60, 26, 32, 36, 28, 42, 48, 36, 60, 74, 40, 8, 52, 60, 52, 132, 40, 46, 114, 14, 72, 48, 36, 120, 96, 72, 60, 64, 60, 100, 124, 60, 68, 126, 52, 60, 124

Offset: 0

Seiichi Manyama, Oct 07 2018

nonn

a(n) > 0 for n >= 0.

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

Cf. A320138, A320139, A320140, A033712, A320188, A320189, A320190.

CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^10], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(20))^5/(eta(q^10)* eta(q^(40)))^2 ) \\ G. C. Greubel, Oct 29 2018

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^10).

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.

A282544 Expansion of (phi(x)^4 + 3*phi(x^3)^4) / 4 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 6, 14, 6, 12, 42, 16, 6, 50, 36, 24, 42, 28, 48, 84, 6, 36, 150, 40, 36, 112, 72, 48, 42, 62, 84, 158, 48, 60, 252, 64, 6, 168, 108, 96, 150, 76, 120, 196, 36, 84, 336, 88, 72, 300, 144, 96, 42, 114, 186, 252, 84, 108, 474, 144, 48, 280, 180, 120, 252

Offset: 0

Michael Somos, Feb 17 2017

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
a(n) is the number of solutions in integers to n = x^2 + y^2 + z^2 + w^2 where x + y + z = 3m is a multiple of 3. - Michael Somos, Jun 23 2018

			G.f. = 1 + 2*x + 6*x^2 + 14*x^3 + 6*x^4 + 12*x^5 + 42*x^6 + 16*x^7 + 6*x^8 + ...
a(4) = 6 with solutions (x, y, z, w) = {(1, 1, 1, 1), (1, 1, 1, -1), (0, 0, 0, 2)} and their negatives. - _Michael Somos_, Jun 23 2018

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A013661, A033712, A125510.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

A := Basis( ModularForms( Gamma0(12), 2), 60); A[1] + 2*A[2] + 6*A[3] + 14*A[4] + 6*A[5];

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[n, # {1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0}[[Mod[#, 12, 1]]] &]];
a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ (Which[# < 3, 2 + (-1)^#, # == 3, 3^(#2 + 1) - 2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger@n)];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^4 + 3 * EllipticTheta[ 3, 0, x^3]^4) / 4, {x, 0, n}];

{a(n) = if( n<1, n==0, 2 * sumdiv(n, d, d * [0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1][d%12+1]))};

{a(n) = if( n<1, n==0, my(A = factor(n), p, e); 2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3, p==3, 3^(e+1) - 2, (p^(e+1) - 1) / (p - 1))))};

{a(n) = if( n<0, 0, my(A); A = x * O(x^n); polcoeff( (sum(k=1, sqrtint(n), 2 * x^k^2, 1 + A)^4 + 3 * sum(k=1, sqrtint(n\3), 2 * x^(3*k^2), 1 + A)^4) / 4, n))};

Expansion of a(x^2) * phi(x) * phi(x^3) in powers of x where a() is a cubic AGM theta function and phi() is a Ramanujan theta function.
Expansion of (chi(x) * chi(x^3))^3 * (psi(x)^4 + 3*x*psi(x^3)^4) in powers of x where psi(), chi() are Ramanujan theta functions.
a(n) = 2*b(n) where b() is multiplicative with a(0) = 1, b(2^e) = 3 if e>0, b(3^e) = 3^(e+1) - 2, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: ((Sum_{k in Z} x^k^2)^4 + 3 * (Sum_{k in Z} x^(3*k^2))^4) / 4.
G.f.: 1 + 2 * Sum_{k>0} F(k, x) + 6 * Sum_{k>0} F(3*k, x) where F(k, x) = x^k / (1 + (-x)^k)^2.
G.f.: 1 + 2 * Sum_{k>0} F(k, x) + 2 * Sum_{k>0} F(3*k, x) where F(k, x) = k * x^k / (1 + (-x)^k).
a(2*n) = A125510(n). a(n) = A033712(2*n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/6 = 1.644934... (A013661). - Amiram Eldar, Dec 29 2023

A028977 Theta series of 8-d 6-modular lattice G_2 tensor F_4 (or A_2 tensor D_4) with det 1296 and minimal norm 4 in powers of q^2.

Original entry on oeis.org

1, 0, 72, 192, 504, 576, 2280, 1728, 4248, 4800, 7920, 6336, 19416, 10368, 21312, 22464, 33624, 24192, 63048, 32832, 65808, 60864, 83232, 57600, 155640, 76032, 137520, 130944, 180288, 116928, 290736

Offset: 0

N. J. A. Sloane

nonn

Proposition 7.6 [McKay and Sebbar, 2000, p. 272, equ. (7.8)] expresses the theta series as a Schwarzian of A007258 and tau. - Michael Somos, Jun 05 2015

			G.f. = 1 + 72*x^2 + 192*x^3 + 504*x^4 + 576*x^5 + 2280*x^6 + 1728*x^7 + ...
G.f. = 1 + 72*q^4 + 192*q^6 + 504*q^8 + 576*q^10 + 2280*q^12 + 1728*q^14 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
G. Nebe and N. J. A. Sloane, Home page for this lattice
E. M. Rains and N. J. A. Sloane, The Shadow Theory of Modular and Unimodular Lattices, J. Number Theory, 73 (1998), 359-389.
Index entries for sequences related to D_4 lattice

Cf. A007258, A030209, A033712, A212770, A212817.

A := Basis( ModularForms( Gamma0(6), 4), 32); A[1] + 72*A[3] + 192*A[4] + 504*A[5]; /* Michael Somos, Aug 20 2014 */

a[ n_] := SeriesCoefficient[ With[{e1 = QPochhammer[ x] QPochhammer[ x^6], e2 = QPochhammer[ x^2] QPochhammer[ x^3]}, (e2^7 / e1^5 - x e1^7 /e2^5)^2 - 8 x (e1 e2)^2], {x, 0, n}]; (* Michael Somos, Apr 19 2015 *)

{a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^3 + A); A = eta(x + A) * eta(x^6 + A); polcoeff( (B^7 / A^5 - x * A^7 / B^5)^2 - 8 * x * (A * B)^2, n))}; /* Michael Somos, May 27 2012 */

Expansion of ((eta(q^2) * eta(q^3))^7 / (eta(q) * eta(q^6))^5 - (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5)^2 - 8 * (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12))^2 in powers of q. - Michael Somos, May 27 2012
A212817(n) = a(n) + 8 * A030209(n). - Michael Somos, May 27 2012
G.f. A(x) = g1(x)^2 * (1 - 4*g2(x) - 16*g2(x)^3 + 16*g2(x)^4) where g1(x) = A033712(x) and g2(x) = A212770(x). - Michael Somos, Apr 19 2015

cp's OEIS Frontend

A320139 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 4*d^2 = n.

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A320138 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 3*d^2 = n.

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A320140 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 5*d^2 = n.

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A320188 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 7*d^2 = n.

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A320189 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 8*d^2 = n.

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A320190 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 9*d^2 = n.

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Mathematica

PARI

Formula

A320191 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 10*d^2 = n.

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Mathematica

PARI

Formula

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

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A320139 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 4*d^2 = n.

A320138 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 3*d^2 = n.

A320140 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 5*d^2 = n.

A320188 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 7*d^2 = n.

A320189 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 8*d^2 = n.

A320190 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 9*d^2 = n.

A320191 Number of integer solutions to a^2 + 2b^2 + 3c^2 + 10*d^2 = n.

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