A007653 -id:A007653 - cp's OEIS frontend

A000748 Expansion of bracket function.

1, -3, 6, -9, 9, 0, -27, 81, -162, 243, -243, 0, 729, -2187, 4374, -6561, 6561, 0, -19683, 59049, -118098, 177147, -177147, 0, 531441, -1594323, 3188646, -4782969, 4782969, 0, -14348907, 43046721, -86093442, 129140163, -129140163, 0, 387420489, -1162261467

Offset: 0

N. J. A. Sloane

It appears that the sequence coincides with its third-order absolute difference. - John W. Layman, Sep 05 2003

It appears that, for n > 0, the (unsigned) a(n) = 3*|A057682(n)| = 3*|Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1)|. - John W. Layman, Sep 05 2003

			G.f. = 1 - 3*x + 6*x^2 - 9*x^3 + 9*x^4 - 27*x^6 + 81*x^7 - 162*x^8 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (-3,-3).

Column 3 of A307047.
Cf. A000749, A000750, A001659.
Cf. A057682.

I:=[1,-3]; [n le 2 select I[n] else -3*Self(n-1)-3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 11 2016

A000748:=(-1-2*z-3*z**2-3*z**3+18*z**5)/(-1+z+9*z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from signs
a:= n-> (Matrix([[ -3,1], [ -3,0]])^n)[1,1]: seq(a(n), n=0..40); # Alois P. Heinz, Sep 06 2008

a[n_] := 2*3^(n/2)*Sin[(1-5*n)*Pi/6]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 12 2014 *)
LinearRecurrence[{-3, -3}, {1, -3}, 40] (* Jean-François Alcover, Feb 11 2016 *)

{a(n) = if( n<0, 0, polcoeff(1 / (1 + 3*x + 3*x^2) + x * O(x^n), n))}; /* Michael Somos, Jun 07 2005 */

{a(n) = if( n<0, 0, 3^((n+1)\2) * (-1)^(n\6) * ((-1)^n + (n%3==2)))}; /* Michael Somos, Sep 29 2007 */

G.f.: 1/((1+x)^3-x^3).
a(n) = A007653(3^n).
a(n) = -3*a(n-1) - 3*a(n-2). - Paul Curtz, May 12 2008
a(n) = Sum_{k=1..n} binomial(k,n-k)*(-3)^(k) for n > 0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
G.f.: 1/(1 + 3*x /(1 - x /(1+x))). - Michael Somos, May 12 2012
G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - 3*x*(2*k+1 + x)/(3*x*(2*k+2 + x) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2014
a(n) = 2*3^(n/2)*sin((1-5*n)*Pi/6). - Jean-François Alcover, Mar 12 2014
a(n) = (-1)^n * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+2,3*k+2). - Seiichi Manyama, Aug 05 2024
a(n) = (i*sqrt(3)/3)*((-3/2 - i*sqrt(3)/2)^(n+1) - (-3/2 + i*sqrt(3)/2)^(n+1)), where i = sqrt(-1). - Taras Goy, Jan 20 2025
a(n) = -2*a(n-1) + 3*a(n-3). - Taras Goy, Jan 26 2025

A130542 The maximum absolute value of the L-series coefficient for an elliptic curve.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 5, 3, 6, 8, 6, 6, 7, 10, 12, 4, 8, 12, 8, 8, 15, 12, 9, 9, 11, 14, 9, 10, 10, 24, 11, 8, 18, 16, 20, 12, 12, 16, 21, 12, 12, 30, 13, 12, 24, 18, 13, 12, 18, 22, 24, 14, 14, 18, 24, 15, 24, 20, 15, 24, 15, 22, 30, 8, 28, 36, 16, 16, 27, 40, 16, 18, 17, 24, 33, 16, 30, 42

Offset: 1

Michael Somos, Jun 04 2007

The values of a(n) and the multiplicativity are conjectural.

Let p be a prime number. By a theorem of Deuring and Waterhouse, for any integer t of absolute value at most floor(2*sqrt(p)), there exists an elliptic curve E having its p-th L-series coefficient as t. This gives the values a(n) for all primes and prime powers n. Multiplicativity of a(n) can be shown by an application of the Chinese remainder theorem for elliptic curves, thus yielding all values of a(n). - Robin Visser, Oct 21 2023

			For example abs(A007653(n)) <= a(n) for all n where A007653 is the L-series for the curve y^2 - y = x^3 - x.

Robin Visser, Table of n, a(n) for n = 1..10000
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

def a(n):
    ans, fcts = 1, Integer(n).factor()
    for pp in fcts:
        max_ap = 1
        for ap in range(-floor(2*sqrt(pp[0])), floor(2*sqrt(pp[0]))+1):
            app = [1, ap]
            for i in range(pp[1]-1): app.append(app[1]*app[-1]-pp[0]*app[-2])
            max_ap = max(max_ap, abs(app[-1]))
        ans *= max_ap
    return ans  # Robin Visser, Oct 21 2023

For primes p : a(p) = floor(2*sqrt(p)) and a(p^2) = floor(2*sqrt(p))^2 - p [Deuring-Waterhouse]. - Robin Visser, Oct 21 2023

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4x^3 + ((m+1)^2 + 8)x^2 - 2(m+3)x + 1 - y^2 = 0.

Original entry on oeis.org

1, -1, -2, 0, -3, -1, 1, 2, 1, 2, -1, -2, -1, -3, 1, 0, 6, 1, 0, 3, 2, 1, -1, -3, 1, -2, -2, -2, -1, 0, -1, 0, -1, 0, -1, 0, 0, 6, -2, 0, 6, -2, 0, 6, -2, 1, 4, 1, 6, -1, 2, 2, 2, 3, -2, -1, -5, 4, 3, 1, -2, -4, -5, -3, -1, 1, 0, -6, -1, 0, -3, -2, 0, -6, -1, 0, -3, -2, 1, -2, -7, 0, 2, -2, -1, 0, -5, -2, -5, 3, 4, -1, 2, 3, -2, 2, 4, 2, -2, 1, 6, -1, 4, 2, 4

Offset: 0

Thomas Scheuerle, Nov 17 2024

Unfortunately, if m is a fraction m = b/c, this triangle can only be used for those coefficients where c and (n+1) are coprime. This is not only because the modulo operation is undefined otherwise, but also because rows of the triangle where (n+1) divides c contain these coefficients with the wrong sign.
The parametrization model for elliptic equations defined by -4*x^3 + ((m+1)^2 + 8)*x^2 - 2*(m+3)*x + 1 - y^2 is also used in A377441. From its relation to Somos-4 sequences, it is known that there is at least one generator point of infinite order if m is an integer > 0 or < -1. If we assume the Birch and Swinnerton-Dyer conjecture to be true, then we expect the associated L-function L(E, s) to be zero at s = 1 for such m.
The relation of m to the J-invariant is given by J(m) = (m^12 + 12*m^11 + 114*m^10 + 628*m^9 + 2823*m^8 + 8184*m^7 + 19036*m^6 + 24552*m^5 + 25407*m^4 + 16956*m^3 + 9234*m^2 + 2916*m + 729)/(m^5 + 4*m^4 + 23*m^3 + 9*m^2) for rational m.
The row sums of the triangle show some connection to the Dedekind psi function (A001615), but will deviate for at least many nonsquarefree n+1.
A short table which shows the Cremona label which corresponds to the L-series obtained for some rational m:
.
  m  | label
 -------------
  -5   655a1
  -4   166a1
  -3   153a1
  -2   58a1
  -1   11a3
 -1/2  26b1
 -1/3  141a1
   1   37a1
   2   158b1
   3   423g1
   4   458a1
   5   1745b1
.

			The triangle T(n, k) begins:
  q^(n+1) 0, 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15     sum  A001615
  --------------------------------------------------------------------
  [q^1]   1                                                   1     1
  [q^2]  -1,-2                                               -3     3
  [q^3]   0,-3,-1                                            -4     4
  [q^4]   1, 2, 1, 2                                          6     6
  [q^5]  -1,-2,-1,-3, 1                                       6     6
  [q^6]   0, 6, 1, 0, 3, 2                                   12    12
  [q^7]   1,-1,-3, 1,-2,-2,-2                                -8     8
  [q^8]  -1, 0,-1, 0,-1, 0,-1, 0                             -4    12 <- not equal
  [q^9]   0, 6,-2, 0, 6,-2, 0, 6,-2                          12    12
  [q^10]  1, 4, 1, 6,-1, 2, 2, 2, 3,-2                       18    18
  [q^11] -1,-5, 4, 3, 1,-2,-4,-5,-3,-1, 1                    12    12
  [q^12]  0,-6,-1, 0,-3,-2, 0,-6,-1, 0,-3,-2                -24    24
  [q^13]  1,-2,-7, 0, 2,-2,-1, 0,-5,-2,-5, 3, 4             -14    14
  [q^14] -1, 2, 3,-2, 2, 4, 2,-2, 1, 6,-1, 4, 2, 4           24    24
  [q^15]  0, 6, 1, 0,-3, 1, 0, 3, 3, 0, 3, 2, 0, 9,-1        24    24
  [q^16]  1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4     24    24

Cf. A001615, A251913, A377441.

Cf. A006571 (main diagonal), A007653 (column 1).

T(n, k) = ellak(ellinit(ellfromeqn(-4*x^3 + ((k+n+2)^2 + 8)*x^2 - 2*(k+n+4)*x + 1 - y^2)),n+1);

T(n, n) = A006571(n), case m =-1. Also the expansion of (eta(q) * eta(q^11))^2 in powers of q.
T(n, 1) = A007653(n), case m = 1.
T(2*n, n) = A251913(2*n+1), case m = -1/2. See first comment.
Let p be an odd prime with good reduction, then T(p-1, k) is odd iff -4*x^3 + ((k+1)^2 + 8)*x^2 - 2*(k+3)*x + 1 == 0 (mod p) has no solution.

cp's OEIS Frontend

A000748 Expansion of bracket function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A130542 The maximum absolute value of the L-series coefficient for an elliptic curve.

Views

Author

Keywords

Comments

Examples

Links

Programs

Sage

Formula

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4x^3 + ((m+1)^2 + 8)x^2 - 2(m+3)x + 1 - y^2 = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A000748 Expansion of bracket function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A130542 The maximum absolute value of the L-series coefficient for an elliptic curve.

Views

Author

Keywords

Comments

Examples

Links

Programs

Sage

Formula

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4*x^3 + ((m+1)^2 + 8)*x^2 - 2*(m+3)*x + 1 - y^2 = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4x^3 + ((m+1)^2 + 8)x^2 - 2(m+3)x + 1 - y^2 = 0.