A028959 -id:A028959 - cp's OEIS frontend

A328532 G.f. = Phi^5*F, where Phi = g.f. for A028930, F = g.f. for A028959.

1, 2, 10, 30, 72, 180, 394, 760, 1474, 2552, 4312, 6764, 10664, 15290, 22764, 31252, 44134, 58360, 80192, 101988, 135852, 169552, 218968, 267170, 340142, 406342, 507626, 600242, 735996, 856054, 1042952, 1193878, 1435032, 1633492, 1943892, 2185304

Offset: 0

N. J. A. Sloane, Oct 18 2019

nonn

Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See p. 196. (A028930 is Phi_1, A028959 is F_1.)

Cf. A028930, A028959.

A328533 G.f. = Phi^4*F^2, where Phi = g.f. for A028930, F = g.f. for A028959.

Original entry on oeis.org

1, 4, 12, 40, 100, 216, 472, 896, 1580, 2732, 4408, 6816, 10400, 15280, 22000, 31088, 43308, 58072, 78804, 102176, 134056, 169408, 218320, 268228, 339768, 409244, 507744, 603368, 738000, 859424, 1041376, 1200664, 1435908, 1639184, 1943224, 2186656

Offset: 0

N. J. A. Sloane, Oct 18 2019

nonn

Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See p. 196. (A028930 is Phi_1, A028959 is F_1.)

Cf. A028930, A028959.

A328534 G.f. = Phi^3*F^3, where Phi = g.f. for A028930, F = g.f. for A028959.

Original entry on oeis.org

1, 6, 18, 50, 132, 276, 530, 1008, 1710, 2760, 4464, 6828, 10268, 15246, 21828, 30820, 42750, 57696, 77032, 101916, 132564, 169736, 216816, 269286, 338490, 412386, 507354, 605278, 739308, 865314, 1042944, 1207962, 1437132, 1643508, 1943628, 2191176

Offset: 0

N. J. A. Sloane, Oct 18 2019

nonn

Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See p. 196. (A028930 is Phi_1, A028959 is F_1.)

Cf. A028930, A028959.

A328535 G.f. = Phi^2*F^4, where Phi = g.f. for A028930, F = g.f. for A028959.

Original entry on oeis.org

1, 8, 28, 68, 160, 344, 608, 1032, 1784, 2836, 4416, 6928, 10348, 15260, 21848, 30112, 42012, 57312, 76308, 101272, 131648, 169064, 215496, 270344, 335484, 415128, 506736, 608908, 738976, 873268, 1045160, 1214468, 1437016, 1648432, 1946832, 2201072

Offset: 0

N. J. A. Sloane, Oct 18 2019

nonn

Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See p. 196. (A028930 is Phi_1, A028959 is F_1.)

Cf. A028930, A028959.

A328536 G.f. = Phi*F^5, where Phi = g.f. for A028930, F = g.f. for A028959.

Original entry on oeis.org

1, 10, 42, 102, 192, 372, 682, 1064, 1706, 2936, 4680, 7084, 10384, 14786, 21564, 30260, 41094, 57288, 76368, 99540, 130972, 167168, 214984, 271402, 333430, 414590, 507546, 613482, 738364, 881518, 1041144, 1225406, 1437184, 1656212, 1951012, 2215480

Offset: 0

N. J. A. Sloane, Oct 18 2019

nonn

Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See p. 196. (A028930 is Phi_1, A028959 is F_1.)

Cf. A028930, A028959.

A030199 Expansion of x * Product_{k>=1} (1 - x^k) * (1 - x^(23*k)).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, -1, 1, 1, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, -1, 1, 1, -1, 0, 0, 0, -1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, -1, 0, 0, -1

Offset: 1

N. J. A. Sloane

Number 40 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - q^2 - q^3 + q^6 + q^8 - q^13 - q^16 + q^23 - q^24 + q^25 + q^26 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. Darmon, Andrew Wiles's Marvelous Proof, Notices Amer. Math. Soc., 64 (No. 3, March 2017), 209-216. See p. 211 equ. 23
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Newform orbit 23.1.b.a.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 434.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A028930, A028959.

Basis( CuspForms( Gamma1(23), 1), 82) [1]; /* Michael Somos, Sep 08 2014 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^23], {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^23 + A), n))}; /* Michael Somos, May 02 2005 */

{a(n) = my(A, p, e, y); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==23, 1, y = sum( x=1, p-1, (x^3 - x - 1)%p == 0); if( y==1, 1-e%2, y, e+1, (e-1)%3 - 1))))}; /* Michael Somos, Oct 19 2005 */

{a(n) = if( n<1, 0, qfrep([2, 1; 1, 12], n, 1)[n] - qfrep([4, 1; 1, 6], n, 1)[n])}; /* Michael Somos, Sep 08 2014 */

{a(n) = if( n<1, 0, mfcoefs(mfeigenbasis(mfinit([23, 1, Mod(22, 23)], 0))[1], n)[n+1])}; /* Michael Somos, Aug 22 2025 */

Expansion of eta(q) * eta(q^23) in powers of q.
Euler transform of period 23 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 2 *u * v * w + 2 * u * w^2 + u^2 * w. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 * u3 * u6 +2 * u1 * u2 * u3 * u6 - 2 * u1 * u6^3 + 2 * u2^2 * u3 * u6 - u2 * u3^3. - Michael Somos, May 02 2005
a(n) is multiplicative with a(23^e) = 1. Let y = number of zeros of x^3 - x - 1 modulo p, then a(p^e) = (1 + (-1)^e)/2 if y = 1, a(p^e) = e+1 if y = 3, a(p^e) = (e-1)%3 - 1 if y = 0. - Michael Somos, Oct 19 2005
a(8*n + 4) = a(23*n + 5) = a(23*n + 7) = a(23*n + 10) = a(23*n + 11) = a(23*n + 14) = a(23*n + 15) = a(23*n + 19) = a(23*n + 20) = a(23*n + 21) = a(23*n + 22) = 0. - Michael Somos, Oct 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (23 t)) = 23^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 08 2014
2 * a(n) = A028959(n) - A028930(n). - Michael Somos, Sep 08 2014

Reference to Martin and Ono added by Chandan Singh Dalawat (dalawat(AT)gmail.com), Jul 23 2010

A028958 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 12 ] (divided by 2).

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 12, 16, 18, 23, 24, 25, 26, 27, 32, 36, 39, 48, 49, 52, 54, 58, 59, 62, 64, 72, 78, 81, 82, 87, 92, 93, 94, 96, 100, 101, 104, 108, 116, 117, 121, 123, 124, 128, 138, 141, 142, 144, 146, 150, 156, 162, 164, 167, 169, 173, 174, 184, 186, 188, 192

Offset: 1

N. J. A. Sloane

nonn

Nonnegative integers of the form x^2 + x*y + 6*y^2, discriminant -23. - Ray Chandler, Jul 12 2014
The Gram matrix is positive-definite, therefore, if w := (1 + sqrt(-23)) / 2, then |x + w*y|^2 = x^2 + x*y + 6*y^2 > 0 for all integers x and y except x = y = 0. - Michael Somos, Mar 28 2015
The theta function of the lattice with basis [1, w] is the g.f. of A028959, therefore, A028959(n) is positive if and only if n is in this sequence. - Michael Somos, Mar 28 2015

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

For primes see A033217. Cf. A028929, A106867.

Cf. A028959.

More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A328093 Expansion of (theta_3(z)theta_3(23z) + theta_2(z)theta_2(23z))^5.

Original entry on oeis.org

1, 10, 40, 80, 90, 112, 260, 480, 700, 1050, 1520, 2160, 2980, 3920, 5920, 8160, 9530, 12800, 16620, 20560, 26672, 30720, 38960, 47690, 52020, 66250, 77380, 87940, 101600, 112720, 134304, 147920, 171020, 185760, 220160, 230400, 263550, 292080, 341200, 346820, 423984, 425680, 516480, 527600, 619120

Offset: 0

N. J. A. Sloane, Oct 17 2019

nonn

Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See F_k, p. 188.

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A028959, A028659, A028660, A328094.

a(n) = polcoeff((1 + 2*x*Ser(qfrep([2, 1; 1, 12], n, 1)))^5, n); \\ Jinyuan Wang, Feb 19 2020

A328094 Expansion of (theta_3(z)theta_3(23z) + theta_2(z)theta_2(23z))^6.

Original entry on oeis.org

1, 12, 60, 160, 252, 312, 568, 1200, 2004, 3036, 4680, 7008, 10264, 14568, 21024, 31280, 42012, 54408, 75284, 99600, 129912, 168688, 210240, 272460, 336048, 404052, 516432, 618224, 736272, 884712, 1033008, 1244976, 1439820, 1666800, 1953288, 2232000, 2548516, 2893848, 3376224, 3756912, 4294344

Offset: 0

N. J. A. Sloane, Oct 17 2019

nonn

Jinyuan Wang, Table of n, a(n) for n = 0..1000
Bülent Köklüce, Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables, The Ramanujan Journal 34.2 (2014): 187-208. See F_6, p. 196.

Cf. A028959, A028659, A028660, A328093.

a(n) = polcoeff((1 + 2*x*Ser(qfrep([2, 1; 1, 12], n, 1)))^6, n); \\ Jinyuan Wang, Feb 20 2020

A224529 Sequence f_n from a paper by Robert Osburn and Brundaban Sahu.

Original entry on oeis.org

1, 2, 6, 26, 142, 876, 5790, 40020, 285582, 2087612, 15551620, 117629724, 900964558, 6973745924, 54464010540, 428645647572, 3396238954446, 27067890450300, 216857021933172, 1745460025192140, 14107695302434356, 114455036696796168, 931738743735004596

Offset: 0

Joerg Arndt, Apr 09 2013

nonn

Conjecture 1.1 of Osburn and Sahu is if p is a prime and JacobiSymbol(p, 23) = 1 and n>0 then a(n * p) == a(n) (mod p). - Michael Somos, Sep 21 2013

			G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 142*x^4 + 876*x^5 + 5790*x^6 + 40020*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Robert Osburn and Brundaban Sahu, Congruences via modular forms, arXiv:0912.0173 [math.NT], (Sep 02 2010).
R. Osburn and B. Sahu, Congruences via modular forms, Proc. Amer. Math. Soc. 139 (2011), 2375-2381.

Cf. A224530 (sequence F_n).

Cf. A028959, A030199.

p := (1+224*x -864*x^2 -544*x^3 +9664*x^4 -26112*x^5 +36288*x^6 -27648*x^7 +9216*x^8) ;
s := (1-14*x+57*x^2-106*x^3+90*x^4-16*x^5-19*x^6)^(1/2) ;
A := (5*(53-400*x+944*x^2-912*x^3+288*x^4)-24*(11-16*x)*s)/p ;
f := 4*x*(1-45*x+865*x^2-9270*x^3+60648*x^4 -249463*x^5+640904*x^6 -987056*x^7 +821224*x^8-249920*x^9 -71232*x^10+20610*x^11 -(1-21*x +148*x^2 -380*x^3+212*x^4)*(1-17*x+90*x^2-142*x^3 -14*x^4)*s)*(6*A)^3/23^6;
ogf := A^(1/4) * hypergeom([1/12, 5/12],[1], f);
series(ogf, x=0, 101);  # Mark van Hoeij, Apr 12 2014

a[ n_] := If[ n < 0, 0, With[ {f = Series[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^23] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^23], {x, 0, n}], g = x QPochhammer[ x] QPochhammer[ x^23]}, SeriesCoefficient[ ComposeSeries[ f, InverseSeries[ g/f ]], {x, 0, n}]]]; (* Michael Somos, Sep 21 2013 *)

n^2 * a(n) = (14*n^2 - 21*n + 9) * a(n-1) + (-57*n^2 + 171*n - 136) * a(n-2) + (106*n^2 - 477*n + 551) * a(n-3) + (-90*n^2 + 540*n - 816) * a(n-4) + (16*n^2 - 120*n + 224) * a(n-5) + (19*n^2 - 171*n + 380) * a(n-6). - Michael Somos, Sep 21 2013

G.f. A(x) satisfies f(q) = A(g(q)) where f is the g.f. for A028959 and g(q) = eta(q) * eta(q^23) / f(q). - Michael Somos, Sep 21 2013

cp's OEIS Frontend

A328532 G.f. = Phi^5*F, where Phi = g.f. for A028930, F = g.f. for A028959.

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A328533 G.f. = Phi^4*F^2, where Phi = g.f. for A028930, F = g.f. for A028959.

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A328534 G.f. = Phi^3*F^3, where Phi = g.f. for A028930, F = g.f. for A028959.

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A328535 G.f. = Phi^2*F^4, where Phi = g.f. for A028930, F = g.f. for A028959.

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A328536 G.f. = Phi*F^5, where Phi = g.f. for A028930, F = g.f. for A028959.

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A030199 Expansion of x * Product_{k>=1} (1 - x^k) * (1 - x^(23*k)).

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Magma

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A028958 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 12 ] (divided by 2).

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Extensions

A328093 Expansion of (theta_3(z)*theta_3(23z) + theta_2(z)*theta_2(23z))^5.

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PARI

A328094 Expansion of (theta_3(z)*theta_3(23z) + theta_2(z)*theta_2(23z))^6.

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Programs

PARI

A224529 Sequence f_n from a paper by Robert Osburn and Brundaban Sahu.

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A328093 Expansion of (theta_3(z)theta_3(23z) + theta_2(z)theta_2(23z))^5.

A328094 Expansion of (theta_3(z)theta_3(23z) + theta_2(z)theta_2(23z))^6.