A320202 -id:A320202 - cp's OEIS frontend

A320193 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 4*d^2 = n.

1, 2, 2, 4, 6, 8, 12, 16, 14, 14, 24, 20, 20, 24, 16, 16, 30, 36, 14, 36, 40, 32, 60, 48, 36, 42, 72, 40, 48, 56, 16, 64, 62, 40, 36, 64, 42, 72, 108, 48, 72, 84, 96, 84, 100, 56, 48, 96, 68, 114, 42, 72, 120, 104, 120, 80, 112, 72, 168, 116, 48, 120, 64, 112, 126, 96

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. A320194, A320195, A320196, A033720, A320197, A320198, A320199, A320200, A320201, A320202.

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

A320194 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 5*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 12, 12, 14, 14, 16, 20, 24, 24, 12, 24, 20, 24, 34, 12, 26, 36, 44, 36, 40, 38, 40, 80, 44, 32, 44, 40, 54, 72, 24, 24, 68, 48, 76, 64, 56, 48, 104, 84, 40, 98, 36, 84, 104, 50, 58, 56, 84, 104, 112, 76, 48, 120, 120, 92, 128, 60, 80, 132, 76, 96

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. A320193, A320195, A320196, A033720, A320197, A320198, A320199, A320200, A320201, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^5).

A320195 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 6*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 10, 12, 10, 14, 16, 12, 24, 28, 12, 20, 28, 16, 26, 36, 8, 24, 48, 24, 30, 42, 28, 40, 72, 28, 32, 60, 34, 60, 64, 24, 28, 76, 36, 56, 80, 40, 60, 84, 72, 52, 96, 48, 72, 86, 42, 80, 56, 52, 82, 96, 36, 72, 112, 60, 120, 124, 60, 84, 124, 56, 96, 132, 32

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. A320193, A320194, A320196, A033720, A320197, A320198, A320199, A320200, A320201, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^6).

A320196 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 7*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 10, 10, 10, 16, 12, 16, 28, 16, 20, 24, 24, 18, 28, 32, 8, 40, 32, 24, 34, 32, 32, 34, 56, 20, 40, 42, 40, 64, 56, 32, 32, 72, 28, 72, 64, 32, 60, 72, 84, 36, 88, 56, 66, 98, 40, 80, 40, 80, 104, 68, 44, 40, 92, 52, 116, 80, 42, 72, 100, 96, 84

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. A320193, A320194, A320195, A033720, A320197, A320198, A320199, A320200, A320201, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^7).

A320197 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 9*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 8, 12, 8, 16, 20, 8, 24, 28, 20, 22, 28, 16, 24, 48, 8, 24, 38, 24, 36, 40, 20, 16, 64, 22, 32, 44, 24, 46, 56, 36, 16, 64, 28, 64, 68, 36, 56, 56, 72, 56, 90, 26, 56, 88, 44, 92, 48, 48, 80, 108, 48, 48, 96, 40, 120, 92, 16, 48, 108, 52

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. A320193, A320194, A320195, A320196, A033720, A320198, A320199, A320200, A320201, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^9).

A320198 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 10*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 10, 8, 12, 20, 8, 16, 28, 24, 22, 24, 24, 16, 40, 32, 8, 30, 32, 32, 36, 28, 16, 32, 54, 32, 32, 36, 28, 52, 56, 16, 26, 44, 32, 60, 48, 44, 40, 72, 84, 46, 74, 32, 72, 76, 48, 80, 24, 64, 92, 72, 52, 52, 96, 64, 92, 84, 32, 92, 96

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. A320193, A320194, A320195, A320196, A033720, A320197, A320199, A320200, A320201, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^10).

A320199 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 11*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 6, 12, 16, 8, 16, 20, 24, 26, 24, 20, 24, 32, 24, 32, 14, 24, 40, 32, 24, 24, 32, 22, 64, 32, 24, 40, 36, 56, 40, 32, 8, 48, 32, 42, 56, 20, 48, 72, 74, 42, 72, 52, 56, 88, 48, 56, 40, 56, 76, 76, 56, 36, 96, 52, 88, 112, 44, 64, 92, 88

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. A320193, A320194, A320195, A320196, A033720, A320197, A320198, A320200, A320201, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^11).

A320200 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 12*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 10, 16, 4, 16, 20, 16, 26, 28, 20, 20, 40, 16, 24, 38, 8, 32, 40, 20, 20, 40, 22, 32, 64, 24, 28, 48, 40, 40, 56, 16, 16, 52, 20, 52, 32, 32, 50, 62, 62, 32, 80, 36, 80, 80, 28, 68, 40, 52, 80, 80, 20, 56, 84, 48, 100, 100, 32, 64

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. A320193, A320194, A320195, A320196, A033720, A320197, A320198, A320199, A320201, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^12).

A320201 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 13*d^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 14, 4, 12, 20, 16, 18, 28, 24, 20, 36, 24, 16, 30, 32, 16, 32, 28, 16, 36, 30, 32, 32, 56, 28, 36, 52, 24, 56, 40, 24, 20, 40, 32, 32, 48, 40, 42, 50, 56, 42, 64, 52, 64, 48, 40, 80, 28, 60, 80, 52, 52, 44, 80, 40, 108, 88, 32, 100

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. A320193, A320194, A320195, A320196, A033720, A320197, A320198, A320199, A320200, A320202.

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4) * theta_3(q^13).

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

A320193 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 4*d^2 = n.

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A320194 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 5*d^2 = n.

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A320195 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 6*d^2 = n.

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A320196 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 7*d^2 = n.

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A320197 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 9*d^2 = n.

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A320198 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 10*d^2 = n.

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A320199 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 11*d^2 = n.

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A320200 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 12*d^2 = n.

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A320201 Number of integer solutions to a^2 + 2*b^2 + 4*c^2 + 13*d^2 = n.

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

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A320193 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 4*d^2 = n.

A320194 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 5*d^2 = n.

A320195 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 6*d^2 = n.

A320196 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 7*d^2 = n.

A320197 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 9*d^2 = n.

A320198 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 10*d^2 = n.

A320199 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 11*d^2 = n.

A320200 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 12*d^2 = n.

A320201 Number of integer solutions to a^2 + 2b^2 + 4c^2 + 13*d^2 = n.

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