login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A319822 Number of solutions to x^2 + 2*y^2 + 5*z^2 + 5*w^2 = n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

REFERENCES

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

LINKS

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.

FORMULA

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.

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

PROG

(PARI) 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

(PARI) 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

(Sage)

Q = DiagonalQuadraticForm(ZZ, [1, 2, 5, 5])

Q.theta_series(68).list() # Peter Luschny, Oct 01 2018

CROSSREFS

Cf. A004018, A033715, A236922-A236933.

From Seiichi Manyama, Oct 07 2018: (Start)

54 possible choice:

  k   |   a, b, c,  d   | Number of solutions

------+-----------------+--------------------

    1 |   1, 1, 1,  1   | A000118

    2 |   1, 1, 1,  2   | A236928

    3 |   1, 1, 1,  3   | A236926

    4 |   1, 1, 1,  4   | A236923

    5 |   1, 1, 1,  5   | A236930

    6 |   1, 1, 1,  6   | A236931

    7 |   1, 1, 1,  7   | A236932

    8 |   1, 1, 2,  2   | A097057

    9 |   1, 1, 2,  3   | A320124

   10 |   1, 1, 2,  4   | A320125

   11 |   1, 1, 2,  5   | A320126

   12 |   1, 1, 2,  6   | A320127

   13 |   1, 1, 2,  7   | A320128

   14 |   1, 1, 2,  8   | A320130

   15 |   1, 1, 2,  9   | A320131

   16 |   1, 1, 2, 10   | A320132

   17 |   1, 1, 2, 11   | A320133

   18 |   1, 1, 2, 12   | A320134

   19 |   1, 1, 2, 13   | A320135

   20 |   1, 1, 2, 14   | A320136

   21 |   1, 1, 3,  3   | A034896

   22 |   1, 1, 3,  4   | A272364

   23 |   1, 1, 3,  5   | A320147

   24 |   1, 1, 3,  6   | A320148

   25 |   1, 2, 2,  2   | A320149

   26 |   1, 2, 2,  3   | A320150

   27 |   1, 2, 2,  4   | A236924

   28 |   1, 2, 2,  5   | A320151

   29 |   1, 2, 2,  6   | A320152

   30 |   1, 2, 2,  7   | A320153

   31 |   1, 2, 3,  3   | A320138

   32 |   1, 2, 3,  4   | A320139

   33 |   1, 2, 3,  5   | A320140

   34 |   1, 2, 3,  6   | A033712

   35 |   1, 2, 3,  7   | A320188

   36 |   1, 2, 3,  8   | A320189

   37 |   1, 2, 3,  9   | A320190

   38 |   1, 2, 3, 10   | A320191

   39 |   1, 2, 4,  4   | A320193

   40 |   1, 2, 4,  5   | A320194

   41 |   1, 2, 4,  6   | A320195

   42 |   1, 2, 4,  7   | A320196

   43 |   1, 2, 4,  8   | A033720

   44 |   1, 2, 4,  9   | A320197

   45 |   1, 2, 4, 10   | A320198

   46 |   1, 2, 4, 11   | A320199

   47 |   1, 2, 4, 12   | A320200

   48 |   1, 2, 4, 13   | A320201

   49 |   1, 2, 4, 14   | A320202

   50 |   1, 2, 5,  6   | A320163

   51 |   1, 2, 5,  7   | A320164

   52 |   1, 2, 5,  8   | A320165

   53 |   1, 2, 5,  9   | A320166

   54 |   1, 2, 5, 10   | A033722

(End)

Sequence in context: A319862 A220977 A054134 * A005127 A140773 A133911

Adjacent sequences:  A319819 A319820 A319821 * A319823 A319824 A319825

KEYWORD

nonn

AUTHOR

Jianing Song, Sep 28 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 17 00:35 EDT 2021. Contains 348048 sequences. (Running on oeis4.)