A168611 -id:A168611 - cp's OEIS frontend

A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

1, -3, -3, 15, -3, -18, 15, -24, -3, 69, -18, -36, 15, -42, -24, 90, -3, -54, 69, -60, -18, 120, -36, -72, 15, -93, -42, 231, -24, -90, 90, -96, -3, 180, -54, -144, 69, -114, -60, 210, -18, -126, 120, -132, -36, 414, -72, -144, 15, -171, -93, 270, -42, -162

Offset: 0

Michael Somos, Jul 30 2007

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 3*q - 3*q^2 + 15*q^3 - 3*q^4 - 18*q^5 + 15*q^6 - 24*q^7 - 3*q^8 +...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).

Cf. A121443, A131944, A134077, A168611.

A := Basis( ModularForms( Gamma0(6), 2), 54); A[1] - 3*A[2] - 3*A[3]; /* Michael Somos, Aug 30 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^2]^3 / (QPochhammer[ q^3] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Nov 21 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -3 Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<1, n==0, -3 * sumdiv(n, d, d * ((d%6==1) + (d%6==5) - 2 * (d%6==3))))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^3 / (eta(x^3 + A) * eta(x^6 + A)), n))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1) )))}; /* Michael Somos, Nov 21 2013 */

A = ModularForms( Gamma0(6), 2, prec=54) . basis();  A[0] - 3*A[1] - 3*A[2]; # Michael Somos, Nov 21 2013

Expansion of eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -3, -6, -2, -6, -3, -4, ...].
a(n) = -3 * b(n) where b() is multiplicative with b(2^e) = 1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - Michael Somos, Nov 21 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 54 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121443.
G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k))).
G.f.: 1 - 3 * (Sum_{k>0} (6*k - 1) * x^(6*k - 1) / (1 - x^(6*k - 1)) - 2*(6*k - 5) * x^(6*k - 3) / (1 - x^(6*k - 3)) + (6*k - 5) * x^(6*k - 5) / (1 -x^(6*k - 5))).
a(n) = a(2*n). a(n) = -3 * A131944(n) unless n=0. a(3^n) = 3 * A168611(n+1). a(2*n + 1) = -3 * A134077(n). - Michael Somos, Nov 21 2013

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Original entry on oeis.org

0, 2, -1, 8, 1, -3, 26, 7, -1, -7, 80, 25, 5, -5, -15, 242, 79, 23, 1, -13, -31, 728, 241, 77, 19, -7, -29, -63, 2186, 727, 239, 73, 11, -23, -61, -127, 6560, 2185, 725, 235, 65, -5, -55, -125, -255, 19682, 6559, 2183, 721, 227, 49, -37, -119, -253, -511, 59048, 19681, 6557, 2179, 713, 211, 17, -101, -247, -509, -1023

Offset: 0

K. G. Stier, Jan 22 2015

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

			Table begins
   0    2   8  26  80..
  -1    1   7  25  79..
  -3   -1   5  23  73..
  -7   -5   1  19  65..
  -15 -13  -7  11  49..
  ..   ..  ..  ..  ..

K. G. Stier, Table of n, a(n) for n = 0..5150

Row 0 (=3^n-1) is A024023.
Row 1 (=3^n-2) is A058481.
Row 2 (=3^n-4) is A168611.
Column 0 (=1-2^n) is (-1)A000225.
Column 1 (=3-2^n) is (-1)A036563.
Column 2 (=9-2^n) is (-1)A185346.
Column 3 (=27-2^n) is (-1)A220087.
0,0-Diagonal (=3^n-2^n) is A001047.
1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.
0,1-Diagonal (=3^n-2^(n+1)) is A003063.
0,2-Diagonal (=3^n-2^(n+2)) is A214091.

Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)

for(i=0, 10, {
     for(j=0, i, print1((3^(i-j)-2^j),", "))
});

cp's OEIS Frontend

A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Sage

Formula

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI