A319822 Number of solutions to x^2 + 2*y^2 + 5*z^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

Author

Jianing Song, Sep 28 2018

Keywords

• nonn

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.

Examples

			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.
		

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.

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)

Programs

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-François Alcover, Jun 15 2019 *)

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

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.