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 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-François 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

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