A097057 -id:A097057 - cp's OEIS frontend

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

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.

A133690 Expansion of (phi(-q) * phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

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

Offset: 0

Michael Somos, Sep 20 2007

sign

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

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004011, A008438, A097057, A097723, A112610, A133657, A133692.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Which[ OddQ[n], DivisorSigma[ 1, n], Mod[n, 4] > 0, -2 DivisorSigma[1, n/2], True, -6 DivisorSum[n/4, # Mod[#, 2] &]]]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n<1, n==0, -4 * if( n%2, sigma(n), n%4, -2 * sigma(n/2), -6 * sumdiv( n/4, d, (d%2)*d )))};

{a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^5 / (eta(x^2 + A)^3 * eta(x^8 + A)^2))^2, n))};

Expansion of (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2 in powers of q.
Euler transform of period 8 sequence [ -4, 2, -4, -8, -4, 2, -4, -4, ...].
a(n) = -4 * b(n) where b() is multiplicative with b(2) = -2, b(2^e) = -6 if e>1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133657.
G.f.: ( Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2 )^2.
a(n) = (-1)^n * A097057(n). Convolution square of A133692.
a(2*n) = 8 * A046897(n) unless n=0. a(2*n + 1) = A008438(n). a(4*n) = A004011(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = -16 * A097723(n).

A111973 Expansion of ((eta(q^2)eta(q^4))^6/(eta(q)eta(q^8))^4-1)/4 in powers of q.

Original entry on oeis.org

1, 2, 4, 6, 6, 8, 8, 6, 13, 12, 12, 24, 14, 16, 24, 6, 18, 26, 20, 36, 32, 24, 24, 24, 31, 28, 40, 48, 30, 48, 32, 6, 48, 36, 48, 78, 38, 40, 56, 36, 42, 64, 44, 72, 78, 48, 48, 24, 57, 62, 72, 84, 54, 80, 72, 48, 80, 60, 60, 144, 62, 64, 104, 6, 84, 96, 68, 108, 96, 96, 72

Offset: 1

Michael Somos, Aug 23 2005

Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373, Entry 31.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.29).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A097057(n)=4*a(n), if n>0.

f[p_, e_] := (p^(e+1)-1)/(p-1); f[2, 1] = 2; f[2, e_] := 6; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)

a(n)=if(n<1, 0, sumdiv(n,d, d*(-1)^((d+1)*(n/d+1))*[2,1,0,1][n/d%4+1]))

{a(n)= local(A); if(n<1, 0, A=x*O(x^n); polcoeff( ((eta(x^2+A)*eta(x^4+A))^6/(eta(x+A)*eta(x^8+A))^4-1)/4, n))}

a(n)= local(x); if(n<1, 0, x=2^valuation(n,2); sigma(n/x)*if(x>2,6,x))

{a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, 2+4*(e>1), (p^(e+1)-1)/(p-1)))))}

Multiplicative with a(2)=2, a(2^e)=6 if e>1, a(p^e)=(p^(e+1)-1)/(p-1) if p>2.
G.f.: ((theta_3(q)theta_3(q^2))^2-1)/4 where theta_3(q)=1+2(q+q^4+q^9+...).
G.f.: Sum_{k>0} 2*x^(4k)/(1+x^(4k))^2 +x^(2k-1)/(1-x^(2k-1))^2 = Sum_{k>0} +(2+(-1)^k)k x^(2k)/(1+x^(2k)) +(2k-1)x^(2k-1)/(1-x^(2k-1)). - Michael Somos, Oct 22 2005

A136028 Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 18, 44, 90, 144, 212, 288, 330, 418, 528, 588, 836, 1008, 1056, 1440, 1386, 1356, 1894, 1644, 2064, 2880, 2484, 3168, 3428, 2838, 3696, 3864, 4128, 5040, 5280, 5760, 5418, 5656, 5988, 5376, 7678, 8208, 7572, 10080, 8208, 7788, 10560, 8652, 10404, 13104

Offset: 0

Michael Somos, Dec 10 2007

nonn

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

			G.f. = 1 + 6*q + 18*q^2 + 44*q^3 + 90*q^4 + 144*q^5 + 212*q^6 + 288*q^7 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A029713, A030207, A033715, A097057.

A := Basis( ModularForms( Gamma1(8), 3), 46);  A[1] + 6*A[2] + 18*A[3] + 44*A[4] + 90*A[5] + 144*A[6] + 212*A[7]; /* Michael Somos, Oct 14 2015 */

nmax=60; CoefficientList[Series[Product[((1-x^k)^2 * (1+x^k)^4 * (1+x^(2*k)) / (1+x^(4*k))^2)^3,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^3, {q, 0, n}]; (* Michael Somos, Oct 14 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) * eta(x^4 + A))^3 / (eta(x + A) * eta(x^8 + A))^2)^3, n))};

Expansion of (eta(q^2) * eta(q^4))^9 / (eta(q) * eta(q^8))^6 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1/(8 t)) = 2^(9/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
G.f.: (Product_{k>0} (1 - x^k)^2 * (1 + x^k)^4 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^3.
a(n) = A029713(n) + 6 * A030207(n). Convolution of A033715 and A097057.
a(n) = A028578(4*n). - Michael Somos, Oct 14 2015

A236922 Number of integer solutions to a^2 + b^2 + 4c^2 + 4d^2 = n.

Original entry on oeis.org

1, 4, 4, 0, 8, 24, 16, 0, 24, 52, 24, 0, 32, 56, 32, 0, 24, 72, 52, 0, 48, 128, 48, 0, 96, 124, 56, 0, 64, 120, 96, 0, 24, 192, 72, 0, 104, 152, 80, 0, 144, 168, 128, 0, 96, 312, 96, 0, 96, 228, 124, 0, 112, 216, 160, 0, 192, 320, 120, 0, 192, 248, 128, 0, 24, 336, 192, 0, 144, 384, 192, 0, 312, 296, 152, 0, 160, 384, 224, 0, 144, 484, 168, 0, 256, 432, 176, 0

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. A097057, A236923.

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 4*s(n/2)
elif (n mod 4) = 3 then 0
else 4*s(n); fi; end;
[seq(f(n),n=1..100)];
# a(0)=1 must be added separately

s[n_] := If[IntegerQ[n], DivisorSigma[1, n], 0]; a[n_] := Which[Mod[n, 4] == 0 , 8*s[n/4]-32*s[n/16], Mod[n, 4] == 2, 4*s[n/2], Mod[n, 4] == 3, 0, True, 4*s[n]]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 06 2014, after Maple *)

See Maple code.

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

A317646 Expansion of (1 + theta_3(q))^2*(1 + theta_3(q^2))^2/16, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 5, 4, 5, 8, 8, 8, 11, 8, 6, 8, 5, 10, 14, 12, 16, 12, 11, 8, 11, 14, 14, 20, 18, 12, 14, 12, 5, 20, 19, 20, 30, 16, 17, 16, 16, 18, 24, 20, 25, 28, 14, 16, 11, 22, 25, 28, 34, 20, 30, 24, 18, 28, 26, 28, 42, 24, 20, 32, 5, 28, 36, 28, 41, 32, 32, 20, 30, 30, 28, 44

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of nonnegative integer solutions to the equation x^2 + y^2 + 2*z^2 + 2*w^2 = n.

			G.f. = 1 + 2*q + 3*q^2 + 4*q^3 + 5*q^4 + 4*q^5 + 5*q^6 + 4*q^7 + 5*q^8 + ...

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A014110, A097057, A317645.

nmax = 75; CoefficientList[Series[(1 + EllipticTheta[3, 0, q])^2 (1 + EllipticTheta[3, 0, q^2])^2/16, {q, 0, nmax}], q]
nmax = 75; CoefficientList[Series[(1 + QPochhammer[-q, -q]/QPochhammer[q, -q])^2 (1 + QPochhammer[-q^2, -q^2]/QPochhammer[q^2, -q^2])^2/16, {q, 0, nmax}], q]

cp's OEIS Frontend

A320134 Number of integer solutions to a^2 + b^2 + 2*c^2 + 12*d^2 = n.

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A320135 Number of integer solutions to a^2 + b^2 + 2*c^2 + 13*d^2 = n.

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A320136 Number of integer solutions to a^2 + b^2 + 2*c^2 + 14*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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A133690 Expansion of (phi(-q) * phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

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A111973 Expansion of ((eta(q^2)eta(q^4))^6/(eta(q)eta(q^8))^4-1)/4 in powers of q.

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A136028 Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.

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A320134 Number of integer solutions to a^2 + b^2 + 2c^2 + 12d^2 = n.

A320135 Number of integer solutions to a^2 + b^2 + 2c^2 + 13d^2 = n.

A320136 Number of integer solutions to a^2 + b^2 + 2c^2 + 14d^2 = n.

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

A236922 Number of integer solutions to a^2 + b^2 + 4c^2 + 4d^2 = n.