This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319822 #97 Jun 15 2019 10:32:59
%S A319822 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,
%T A319822 40,56,64,72,8,48,66,24,68,8,46,88,60,32,4,52,64,116,76,12,64,72,60,
%U A319822 82,26,72,104,104,88,8,112,56,136,140,8,136,96,72,98,16,72,132
%N A319822 Number of solutions to x^2 + 2*y^2 + 5*z^2 + 5*w^2 = n.
%C A319822 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.
%C A319822 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.
%D A319822 J. H. Conway, Universal quadratic forms and the fifteen theorem, Contemporary Mathematics 272 (1999), 23-26.
%H A319822 Seiichi Manyama, <a href="/A319822/b319822.txt">Table of n, a(n) for n = 0..10000</a>
%H A319822 L. E. Dickson, <a href="https://doi.org/10.1090/S0002-9904-1927-04312-9">Integers represented by positive ternary quadratic forms</a>, Bulletin of the American Mathematical Society, 1927, 33(1):63-70.
%H A319822 H. D. Kloosterman, <a href="https://projecteuclid.org/euclid.acta/1487102066">On the representation of numbers in the form ax^2 + by^2 + cz^2 + dt^2</a>, Acta Mathematica, 1927, 49(3-4):407-464.
%H A319822 S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram20.pdf">On the expression of a number in the form ax^2 + by^2 + cz^2 + du^2</a>, Proc. Camb. Phil. Soc. 19 (1917), 11-21.
%F A319822 a(n) = Sum_{k=0..floor(n/5)} A004018(k)*A033715(n-5*k).
%F A319822 G.f.: theta_3(q)*theta_3(q^2)*theta_3(q^5)^2, where theta_3() is the Jacobi theta function.
%e A319822 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.
%p A319822 JT := (k, n) -> JacobiTheta3(0, x^k)^n:
%p A319822 A319822List := proc(len) series(JT(1,1)*JT(2,1)*JT(5,2), x, len+1);
%p A319822 seq(coeff(%, x, j), j=0..len) end: A319822List(67); # _Peter Luschny_, Oct 01 2018
%t A319822 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 *)
%o A319822 (PARI) A004018(n) = if(n, 4*sumdiv(n,d,kronecker(-4,d)), 1);
%o A319822 A033715(n) = if(n, 2*sumdiv(n,d,kronecker(-2,d)), 1);
%o A319822 a(n) = my(i=0); for(k=0, n\5, i+=A004018(k)*A033715(n-5*k)); i
%o A319822 (PARI) N=99; q='q+O('q^N);
%o A319822 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;
%o A319822 Vec(gf) \\ _Altug Alkan_, Oct 01 2018
%o A319822 (Sage)
%o A319822 Q = DiagonalQuadraticForm(ZZ, [1, 2, 5, 5])
%o A319822 Q.theta_series(68).list() # _Peter Luschny_, Oct 01 2018
%Y A319822 Cf. A004018, A033715, A236922-A236933.
%Y A319822 From _Seiichi Manyama_, Oct 07 2018: (Start)
%Y A319822 54 possible choice:
%Y A319822 k | a, b, c, d | Number of solutions
%Y A319822 ------+-----------------+--------------------
%Y A319822 1 | 1, 1, 1, 1 | A000118
%Y A319822 2 | 1, 1, 1, 2 | A236928
%Y A319822 3 | 1, 1, 1, 3 | A236926
%Y A319822 4 | 1, 1, 1, 4 | A236923
%Y A319822 5 | 1, 1, 1, 5 | A236930
%Y A319822 6 | 1, 1, 1, 6 | A236931
%Y A319822 7 | 1, 1, 1, 7 | A236932
%Y A319822 8 | 1, 1, 2, 2 | A097057
%Y A319822 9 | 1, 1, 2, 3 | A320124
%Y A319822 10 | 1, 1, 2, 4 | A320125
%Y A319822 11 | 1, 1, 2, 5 | A320126
%Y A319822 12 | 1, 1, 2, 6 | A320127
%Y A319822 13 | 1, 1, 2, 7 | A320128
%Y A319822 14 | 1, 1, 2, 8 | A320130
%Y A319822 15 | 1, 1, 2, 9 | A320131
%Y A319822 16 | 1, 1, 2, 10 | A320132
%Y A319822 17 | 1, 1, 2, 11 | A320133
%Y A319822 18 | 1, 1, 2, 12 | A320134
%Y A319822 19 | 1, 1, 2, 13 | A320135
%Y A319822 20 | 1, 1, 2, 14 | A320136
%Y A319822 21 | 1, 1, 3, 3 | A034896
%Y A319822 22 | 1, 1, 3, 4 | A272364
%Y A319822 23 | 1, 1, 3, 5 | A320147
%Y A319822 24 | 1, 1, 3, 6 | A320148
%Y A319822 25 | 1, 2, 2, 2 | A320149
%Y A319822 26 | 1, 2, 2, 3 | A320150
%Y A319822 27 | 1, 2, 2, 4 | A236924
%Y A319822 28 | 1, 2, 2, 5 | A320151
%Y A319822 29 | 1, 2, 2, 6 | A320152
%Y A319822 30 | 1, 2, 2, 7 | A320153
%Y A319822 31 | 1, 2, 3, 3 | A320138
%Y A319822 32 | 1, 2, 3, 4 | A320139
%Y A319822 33 | 1, 2, 3, 5 | A320140
%Y A319822 34 | 1, 2, 3, 6 | A033712
%Y A319822 35 | 1, 2, 3, 7 | A320188
%Y A319822 36 | 1, 2, 3, 8 | A320189
%Y A319822 37 | 1, 2, 3, 9 | A320190
%Y A319822 38 | 1, 2, 3, 10 | A320191
%Y A319822 39 | 1, 2, 4, 4 | A320193
%Y A319822 40 | 1, 2, 4, 5 | A320194
%Y A319822 41 | 1, 2, 4, 6 | A320195
%Y A319822 42 | 1, 2, 4, 7 | A320196
%Y A319822 43 | 1, 2, 4, 8 | A033720
%Y A319822 44 | 1, 2, 4, 9 | A320197
%Y A319822 45 | 1, 2, 4, 10 | A320198
%Y A319822 46 | 1, 2, 4, 11 | A320199
%Y A319822 47 | 1, 2, 4, 12 | A320200
%Y A319822 48 | 1, 2, 4, 13 | A320201
%Y A319822 49 | 1, 2, 4, 14 | A320202
%Y A319822 50 | 1, 2, 5, 6 | A320163
%Y A319822 51 | 1, 2, 5, 7 | A320164
%Y A319822 52 | 1, 2, 5, 8 | A320165
%Y A319822 53 | 1, 2, 5, 9 | A320166
%Y A319822 54 | 1, 2, 5, 10 | A033722
%Y A319822 (End)
%K A319822 nonn
%O A319822 0,2
%A A319822 _Jianing Song_, Sep 28 2018