cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A006571 Expansion of q*Product_{k>=1} (1-q^k)^2*(1-q^(11*k))^2.

Original entry on oeis.org

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2, 2, -2, -1, 0, -4, -8, 5, -4, 0, 2, 7, 8, -1, 4, -2, -4, 3, 0, -4, 0, -8, -4, -6, 2, -2, 2, 8, 4, -3, 8, 2, 8, -6, -10, 1, 0, 0, 0, 5, -2, 12, -14, 4, -8, 4, 2, -7, -4, 1, 4, -3, 0, 4, -6, 4, 0, -2, 8, -10, -4, 1, 16, -6, 4, -2, 12, 0, 0, 15, 4, -8, -2, -7, -16, 0, -8, -7, 6, -2, -8
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number 23 of the 74 eta-quotients listed in Table I of Martin (1996).
Unique cusp form of weight 2 for congruence group Gamma_1(11). - Michael Somos, Aug 11 2011
For some elliptic curves with p-defects given by this sequence, and for more references, see A272196. See also the Michael Somos formula from May 23 2008 below. - Wolfdieter Lang, Apr 25 2016

Examples

			G.f.: q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + ...

References

  • Barry Cipra, What's Happening in the Mathematical Sciences, Vol. 5, Amer. Math. Soc., 2002; see p. 5.
  • M. du Sautoy, Review of "Love and Math: The Heart of Hidden Reality" by Edward Frenkel, Nature, 502 (Oct 03 2013), p. 36.
  • N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 42.
  • J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 412.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • A. Wiles, Modular forms, elliptic curves and Fermat's last theorem, pp. 243-245 of Proc. Intern. Congr. Math. (Zurich), Vol. 1, 1994.

Links

Crossrefs

Cf. A002070 (terms with prime indices), A032442, A030200.

Programs

  • Magma
    [ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1),n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 29 2007 */
  • Magma
    [ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 31 2007 */
  • Magma
    Basis( CuspForms( Gamma1(11), 2), 101) [1]; /* Michael Somos, Jul 14 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ q (Product[ (1 - q^k), {k, 11, n, 11}] Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
  • PARI
    {a(n) = if( n<1, 0, ellak( ellinit( [0, -1, 1, 0, 0], 1), n))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};
  • PARI
    {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, 1, a0=1; a1 = y = -sum(x=0, p-1, kronecker( 4*x^3 - 4*x^2 + 1, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */
  • Sage
    CuspForms( Gamma1(11), 2, prec = 101).0 # Michael Somos, Aug 11 2011

Formula

Expansion of (eta(q) * eta(q^11))^2 in powers of q.
a(n) == A000594(n) (mod 11). [Cowles]. - R. J. Mathar, Feb 13 2007
Euler transform of period 11 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...]. - Michael Somos, Feb 12 2006
a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) for p != 11. - Michael Somos, Feb 12 2006
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u*w * (u + 4*v + 4*w) - v^3. - Michael Somos, Mar 21 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11 (t/i)^2 f(t) where q = exp(2 Pi i t).
Convolution square of A030200.
Coefficients of L-series for elliptic curve "11a3": y^2 + y = x^3 - x^2. - Michael Somos, May 23 2008
Convolution inverse is A032442. - Michael Somos, Apr 21 2015
a(prime(n)) = prime(n) - A272196(n), n >= 3.
a(2) = -2 is not 2 - A272196(1) = 0. Modularity pattern of some elliptic curves. - Wolfdieter Lang, Apr 25 2016

A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Original entry on oeis.org

1, 0, 4, 5, 0, 4, 0, 8, 0, 8, 21, 0, 0, 0, 4, 0, 20, 0, 0, 0, 16, 40, 0, 0, 0, 0, 0, 9, 0, 32, 0, 16, 0, 0, 0, 16, 45, 0, 24, 0, 0, 0, 0, 0, 12, 0, 84, 0, 0, 0, 0, 0, 0, 0, 16, 111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32
Offset: 1

Views

Author

Mats Granvik, Oct 08 2023

Keywords

Comments

Row n appears to have sum n^2. T(prime(m),1) = A366346(m). The number of nonzero terms in row n appears to be A320111(n).

Examples

			{
{1}, = 1^2
{0, 4}, = 2^2
{5, 0, 4}, = 3^2
{0, 8, 0, 8}, = 4^2
{21, 0, 0, 0, 4}, = 5^2
{0, 20, 0, 0, 0, 16}, = 6^2
{40, 0, 0, 0, 0, 0, 9}, = 7^2
{0, 32, 0, 16, 0, 0, 0, 16}, = 8^2
{45, 0, 24, 0, 0, 0, 0, 0, 12}, = 9^2
{0, 84, 0, 0, 0, 0, 0, 0, 0, 16}, = 10^2
{111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 11^2
{0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32} = 12^2
}

Crossrefs

Cf. A127649, A366346, A000290, A060457, A002070, A036689, A001248, A272196.

Programs

  • Mathematica
    f = x^3 - x^2 - y^2 - y; nn = 12; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.
Conjecture: T(n,n) = A060457(n).

A185699 Expansion of (11 * E_2(x^11) - E_2(x)) / 2 in powers of x where E_2() is an Eisenstein series.

Original entry on oeis.org

5, 12, 36, 48, 84, 72, 144, 96, 180, 156, 216, 12, 336, 168, 288, 288, 372, 216, 468, 240, 504, 384, 36, 288, 720, 372, 504, 480, 672, 360, 864, 384, 756, 48, 648, 576, 1092, 456, 720, 672, 1080, 504, 1152, 528, 84, 936, 864, 576, 1488, 684, 1116, 864, 1176, 648
Offset: 0

Views

Author

Michael Somos, Feb 10 2011

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 5 + 12*x + 36*x^2 + 48*x^3 + 84*x^4 + 72*x^5 + 144*x^6 + 96*x^7 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 480, Entry 8(i).
  • Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, p. 246 (corrected).

Links

Crossrefs

Cf. A006352, A006571, A272196.

Programs

  • Mathematica
    terms = 54;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
(11*E2[x^11] - E2[x])/2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
  • PARI
    {a(n) = if( n<1, 5 * (n==0), 12 * (sigma( n) - if( n%11, 0, 11 * sigma( n / 11))))};

Formula

Expansion of 5 * (phi(x) * phi(x^11))^2 - 20 * x * (f(x) * f(x^11))^2 + 32 * x^2 * (f(-x^2) * f(-x^22))^2 - 20 * x^3 * (psi(-x) * psi(-x^11))^2 in powers of x where f(), phi(), psi() are Ramanujan theta functions.
Expansion of 5 + 12*Sum_{n>=1} Chi0(n)*n*q^n / (1 - q^n), where Chi0(n) = 1 if gcd(n,11) = 1 and 0 otherwise. See the Moreno-Wagstaff reference p. 246, second equation multiplied by 12 (a misprint has been corrected, after mail exchange with C. J. Moreno). - Wolfdieter Lang, Jan 02 2017

A366346 a(n) = A002070(n) + A036689(n).

Original entry on oeis.org

0, 5, 21, 40, 111, 160, 270, 342, 505, 812, 937, 1335, 1632, 1800, 2170, 2750, 3427, 3672, 4415, 4967, 5260, 6152, 6800, 7847, 9305, 10102, 10490, 11360, 11782, 12665, 16010, 17012, 18625, 19192, 22042, 22652, 24485, 26410, 27710, 29750, 31847, 32587, 36307
Offset: 1

Views

Author

Mats Granvik, Oct 07 2023

Keywords

Crossrefs

Cf. A366362, A002070, A036689, A001248, A272196.

Programs

  • Mathematica
    b[n_] := If[n < 1, 0, With[{m = Prime@n}, SeriesCoefficient[q (Product[(1 - q^(11 k)), {k, Ceiling[m/11]}] Product[1 - q^k, {k, m}])^2, {q, 0, m}]]]; Table[Prime[n] (Prime[n] - 1) + b[n], {n, 1, 43}] (* after Michael Somos in A002070, Jul 04 2011 *)

Formula

a(n) = A002070(n) + A036689(n).
a(n) = Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} [GCD(f(x,y), prime(n)) = 1],
a(n) = Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} (1 - [MOD(f(x,y), prime(n)) = 0]) where f(x,y) = x^3 - x^2 - y^2 - y, in the last two formulas.
a(n) = A001248(n) - A272196(n), for n > 1.
a(n) = A366362(prime(n), 1).

A376073 Number of solutions of the congruence y^2 + y == x^3 - x^2 (mod p) as p runs through the primes.

Original entry on oeis.org

4, 4, 4, 9, 10, 9, 19, 19, 24, 29, 24, 34, 49, 49, 39, 59, 54, 49, 74, 74, 69, 89, 89, 74, 104, 99, 119, 89, 99, 104, 119, 149, 144, 129, 159, 149, 164, 159, 179, 179, 194, 174, 174, 189, 199, 199, 199, 204, 209, 214, 209, 269, 249, 274, 259, 249, 259, 299, 279, 299
Offset: 1

Views

Author

Andrey Zabolotskiy, Sep 08 2024

Keywords

Comments

Same as A272196, except for a(1).

References

  • Edward Frenkel, Love and math: the heart of hidden reality, Basic Books, 2013. See pages 86-89.

Links

Crossrefs

Cf. A002070, A006571, A060457, A272196.

Formula

a(n) = prime(n) - A002070(n).
a(n) = A060457(prime(n)).
Showing 1-5 of 5 results.

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