A033716 -id:A033716 - cp's OEIS frontend

A115979 Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.

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

Offset: 1

Michael Somos, Feb 09 2006

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033716, A096936, A115978.

S:=series((1-JacobiTheta4(0,q)*JacobiTheta4(0,q^3))/2, q, 106):
seq(coeff(S,q,n),n=1..105); # Robert Israel, Nov 20 2017

Drop[CoefficientList[Series[(1 -EllipticTheta[4, 0, q]*EllipticTheta[4, 0, q^3])/2, {q, 0, 110}], q], 1] (* G. C. Greubel, May 09 2019 *)

{a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^3+A))^2/eta(x^2+A)/eta(x^6+A), n)/-2)}

def E(x): return 1 + 2*sum((-1)^k*x^(k^2) for k in (1..50))
a=((1 - E(x)*E(x^3))/2).series(x, 110).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 09 2019

(define (A115979 n) (- (* (expt -1 n) (A096936 n)))) ;; Follow A096936 for the rest of code. - Antti Karttunen, Nov 20 2017

Expansion of (1-(eta(q)*eta(q^3))^2/(eta(q^2)*eta(q^6)))/2 in powers of q.
Moebius transform is period 12 sequence [1,-1,0,-3,-1,0,1,3,0,1,-1,0,...].
a(n) is multiplicative and a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e)=1, a(p^e) = 1+e if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} x^(k)/(1+x^k+x^(2k)) -4x^(4k)/(1+x^(4k)+x^(8k)).
a(n) = -(-1)^n*A096936(n).
A115978(n) = -2*a(n) if n > 0.

A228447 Expansion of q * (psi(q^3) * psi(q^6)) / (psi(q) * phi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 7, -15, 30, -57, 104, -183, 313, -522, 852, -1365, 2150, -3336, 5106, -7719, 11538, -17067, 25004, -36306, 52280, -74700, 105960, -149277, 208951, -290706, 402127, -553224, 757158, -1031166, 1397744, -1886151, 2534316, -3391254, 4520112, -6002007

Offset: 1

Michael Somos, Oct 26 2013

sign

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

			G.f. = q - 3*q^2 + 7*q^3 - 15*q^4 + 30*q^5 - 57*q^6 + 104*q^7 - 183*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033716, A093829, A128638, A187100, A187145.

a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^(3/2)]^3 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3]), {q, 0, n}]

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^7 * eta(x^3 + A)), n))}

Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(q) * phi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q)^3 * eta(x^4)^2 * eta(x^6) * eta(x^12)^2 / (eta(x^2)^7 * eta(x^3)) in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - 3*v) - v * (1 - 4*v) * (1 - 3*u)^2.
a(n) = -(-1)^n * A187100(n). a(2*n) = -3 * A128638(n).
Convolution inverse is A187145. Convolution with A033716 is A093829.

A129576 Expansion of phi(x) * chi(x) * psi(-x^3) in powers of x where phi(), chi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 2, 0, 2, 3, 2, 0, 1, 6, 2, 0, 2, 0, 2, 0, 3, 6, 0, 0, 2, 3, 2, 0, 2, 6, 2, 0, 0, 0, 4, 0, 2, 3, 2, 0, 2, 6, 0, 0, 1, 6, 2, 0, 4, 0, 2, 0, 0, 6, 2, 0, 2, 0, 2, 0, 3, 6, 2, 0, 2, 0, 0, 0, 2, 9, 2, 0, 0, 6, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 6, 4, 0, 0, 3, 4

Offset: 0

Michael Somos, Apr 23 2007

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). - Michael Somos, Jun 28 2017

			G.f. = 1 + 3*x + 2*x^2 + 2*x^4 + 3*x^5 + 2*x^6 + x^8 + 6*x^9 + 2*x^10 + ...
G.f. = q + 3*q^4 + 2*q^7 + 2*q^13 + 3*q^16 + 2*q^19 + q^25 + 6*q^28 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033716, A093602, A096936, A112298, A122861.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := Length @ FindInstance[ x^2 + 3 y^2 == 3 n + 1, {x, y}, Integers, 10^9] / 2; (* Michael Somos, Sep 03 2016 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# == 3, Boole[#2 == 0], # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 2, (1 + (-1)^#2)/2, True, #2 + 1] & @@@ FactorInteger[3 n + 1])]; (* Michael Somos, Jun 28 2017 *)

{a(n) = if( n<0, 0, n = 3*n + 1; sumdiv(n, d, kronecker(-3, d) * [0, 1, -2, 1] [n/d%4 + 1] ))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};

{a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3*(1-e%2), p==3, 0, p%3==2, 1-e%2, e+1)))}; /* Michael Somos, Jun 28 2017 */

From Michael Somos, Jun 28 2017: (Start)
Expansion of q^(-1/3) * (2*c(q) + c(-q)) / 3 = q^(-1/3) * (c(q) + 2*c(q^4)) / 3 in powers of q where c() is a cubic AGM theta function.
Expansion of (a(q) - a(q^3) + 2*a(q^4) - 2*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q^(-1/3) * eta(q^2)^7 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)) in powers of q. (End)
Euler transform of period 12 sequence [3, -4, 2, -1, 3, -4, 3, -1, 2, -4, 3, -2, ...].
a(n) = b(3*n + 1) where b() is multiplicative and b(2^e) = 3 * (1 + (-1)^e) / 2 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1 + (-1)^e)/2 if p == 2 (mod 3).
a(n) = A096936(3*n + 1) = A112298(3*n + 1).
2 * a(n) = A033716(3*n + 1). - Michael Somos, Sep 03 2016
a(n) = (-1)^n * A122161(n). - Michael Somos, Jun 28 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Dec 29 2023

A239074 Given a circle of radius R into which small circles of radius R/2^n are packed in a "hexagonal pattern" (see Comments), a(n) is the number of points at which a small circle is tangent to the big circle.

Original entry on oeis.org

2, 2, 6, 2, 6, 6, 6, 2, 18, 6, 2, 18, 6, 18, 54, 2, 6, 54, 6, 6, 90, 2, 2, 54, 54, 18, 54, 18, 6, 162, 6, 2, 18, 6, 18, 1458, 18, 18, 162, 18, 2, 810, 6, 18, 1458, 2, 6, 486, 18, 162, 486, 54, 6, 486, 18, 54, 162, 18, 2, 4374, 6, 18, 2430, 6, 54, 162, 18, 18, 18, 54

Offset: 1

Kival Ngaokrajang, Apr 06 2014

nonn

The construction rule is: (1) Start with a unit circle (big R-circle). (2) Pack circles at radius 1/2^n (small r-circles) on the diameter line of the big circle. (3) Pack small circles in the rows above and below the row packed in the previous step, maintaining a hexagonal packing pattern. The number of small circles in any row is limited so that the circumference of the last small circle does not cross (but is allowed to contact) the circumference of the big circle. (4) Repeat process to the top and bottom rows.
The contact points are the points where the circumference of a small circle contacts the circumference of the big circle, i.e., they are mutually tangent.
See illustration in links.
Also, the number of integer solutions to the equation (2^n-1)^2 = 3*x^2 + y^2. - Andrew Howroyd, May 27 2018

			n=9 (see the link): In the first quadrant, shown there, there are 4 touching points with the large circle for rows x > 0, namely for the rows 52, 132, 280 and 292. With the trivial 2 touching points with the large circle for the row x=0 this adds to the total number 2 + 4*4 = 18 = a(9). - _Wolfdieter Lang_, Apr 06 2014

Kival Ngaokrajang, Illustration of initial terms
Kival Ngaokrajang, Illustration for n = 9

Cf. A033716, A239073, A239206.

a[1] = 2; a[n_] := Module[{f = FactorInteger[2^n - 1]}, 2*Product[If[Mod[ f[[i, 1]], 3] == 1, 2*f[[i, 2]] + 1, 1] , {i, 1, Length[f]}]];
Array[a, 70] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

a(n) = {my(f=factor(2^n-1)); 2*prod(i=1, #f~, if(f[i, 1]%3==1, 2*f[i, 2]+1, 1))} \\ Andrew Howroyd, May 27 2018

a(n) = 2 + 4*A(n), n >= 1, with A(n) the number of integer solutions for x(n,j) = sqrt((2^n-1)^2 + 3*j^2), for j = 1, 2, ..., floor((2^n-1)/sqrt(3)). R = 2^n and r=1 (small radius) was used here. - Wolfdieter Lang, Apr 07 2014

a(n) = A033716((2^n - 1)^2). - Andrew Howroyd, May 27 2018

Corrected and extended by Wolfdieter Lang, Apr 06 2014

a(26)-a(70) from Andrew Howroyd, May 27 2018

A320239 Expansion of theta_3(q) * theta_3(q^3) * theta_3(q^5), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 0, 2, 6, 2, 4, 4, 4, 14, 0, 0, 14, 4, 4, 0, 6, 12, 8, 4, 2, 20, 0, 4, 20, 2, 8, 10, 12, 4, 4, 4, 16, 32, 0, 0, 26, 4, 0, 12, 0, 20, 8, 4, 8, 6, 4, 4, 42, 18, 0, 8, 20, 12, 16, 0, 12, 48, 8, 8, 0, 16, 8, 12, 14, 0, 16, 4, 20, 24, 4, 0, 36, 28, 0, 2, 20, 8, 8, 4, 6

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, a_3) to the equation a_1^2 + 3*a_2^2 + 5*a_3^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^(2*k-1)): A000122 (m=1), A033716 (m=2), this sequence (m=3), A320240 (m=4).

Cf. A320078.

A320240 Expansion of theta_3(q) * theta_3(q^3) * theta_3(q^5) * theta_3(q^7), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 0, 2, 6, 2, 4, 6, 8, 14, 4, 12, 18, 12, 12, 8, 34, 12, 8, 32, 10, 28, 0, 16, 44, 18, 16, 14, 54, 8, 12, 48, 32, 52, 28, 32, 42, 40, 8, 44, 92, 28, 16, 56, 28, 30, 44, 12, 86, 74, 8, 32, 72, 24, 40, 104, 72, 56, 32, 56, 56, 112, 8, 38, 166, 24, 36, 40, 56, 88, 52

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, a_3, a_4) to the equation a_1^2 + 3*a_2^2 + 5*a_3^2 + 7*a_4^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^(2*k-1)): A000122 (m=1), A033716 (m=2), A320239 (m=3), this sequence (m=4).

Cf. A320078.

A374295 a(n) is the smallest positive integer k such that A096936(k) = n.

Original entry on oeis.org

1, 7, 4, 91, 2401, 28, 117649, 1729, 196, 31213, 282475249, 364, 13841287201, 1529437, 9604, 53599, 33232930569601, 2548, 1628413597910449, 593047, 470596, 3672178237, 3909821048582988049, 6916, 68574961, 179936733613, 33124, 29059303, 459986536544739960976801, 124852

Offset: 1

Seiichi Manyama, Jul 02 2024

nonn

a(n) is the smallest positive integer k such that A033716(k) = 2*n,

			   n        |        a(n)
------------+-------------------------------------
   2        |            7.
   3 = 3*1  |            4.
   4        |           91 =     7 * 13.
   5        |         2401 =     7^4.
   6 = 3*2  |           28 = 4 * 7.
   7        |       117649 =     7^6.
   8        |         1729 =     7 * 13 * 19.
   9 = 3*3  |          196 = 4 * 7^2.
  10        |        31213 =     7^4 * 13.
  11        |    282475249 =     7^10.
  12 = 3*4  |          364 = 4 * 7 * 13.
  13        |  13841287201 =     7^12.
  14        |      1529437 =     7^6 * 13.
  15 = 3*5  |         9604 = 4 * 7^4.
  16        |        53599 =     7 * 13 * 19 * 31.
  17        |                    7^16.
  18 = 3*6  |         2548 = 4 * 7^2 * 13.
  19        |                    7^18.
  20        |       593047 =     7^4 * 13 * 19.
  21 = 3*7  |       470596 = 4 * 7^6.
  22        |   3672178237 =     7^10 * 13.
  23        |                    7^22.
  24 = 3*8  |         6916 = 4 * 7 * 13 * 19.
  25        |     68574961 =     7^4 * 13^4.
  26        | 179936733613 =     7^12 * 13.
  27 = 3*9  |        33124 = 4 * 7^2 * 13^2.
  28        |     29059303 =     7^6 * 13 * 19.
  29        |                    7^28.
  30 = 3*10 |       124852 = 4 * 7^4 * 13.

Cf. A033716, A096936, A343771.

If p is prime, a(p) = 7^(p-1).

a(n) is divisible by 7 for n > 3.

A028613 Expansion of theta_3(q) * theta_3(q^12) + theta_2(q) * theta_2(q^12) in powers of q^(1/4).

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 2*x^4 + 4*x^13 + 2*x^16 + 4*x^21 + 2*x^36 + 4*x^37 + 2*x^48 + ...
G.f. = 1 + 2*q + 4*q^(13/4) + 2*q^4 + 4*q^(21/4) + 2*q^9 + 4*q^(37/4) + 2*q^12 + ...

Cf. A033716, A112604, A112607.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^12] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^12], {x, 0, n/4}]; (* Michael Somos, Feb 22 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^96+A))^5 / (eta(x^4 + A) * eta(x^16 + A) * eta(x^48 + A) * eta(x^192 + A))^2 + 4*x^13 * (eta(x^16 + A) * eta(x^192 + A))^2 / (eta(x^8 + A) * eta(x^96 + A)), n))};

a(4*n + 2) = a(4*n + 3) = a(8*n + 1) = a(16*n + 8) = a(16*n + 12) = 0. - Michael Somos, Feb 22 2015

a(8*n + 5) = 4*A112607(n-1). a(16*n) = A033716(n). a(16*n + 4) = 2*A112604(n). - Michael Somos, Feb 22 2015

A306518 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{d|k} theta_3(q^d).

Original entry on oeis.org

1, 1, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 0, 4, 2, 1, 2, 2, 2, 2, 0, 1, 2, 0, 4, 6, 0, 0, 1, 2, 2, 0, 4, 0, 4, 0, 1, 2, 0, 6, 2, 4, 0, 0, 0, 1, 2, 2, 0, 6, 2, 8, 4, 2, 2, 1, 2, 0, 4, 2, 4, 4, 8, 0, 6, 0, 1, 2, 2, 2, 4, 0, 14, 0, 6, 2, 0, 0, 1, 2, 0, 4, 6, 4, 0, 8, 0, 6, 0, 4, 0, 1, 2, 2, 0, 2, 0, 8, 2, 6, 6, 8, 0, 4, 0

Offset: 0

Ilya Gutkovskiy, Feb 21 2019

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  2,  2,  2,  2,  2,  2,  ...
  0,  2,  0,  2,  0,  2,  ...
  0,  4,  2,  4,  0,  6,  ...
  2,  2,  6,  4,  2,  6,  ...
  0,  0,  0,  4,  2,  4,  ...

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Columns k=1..48 give A000122, A033715, A033716, A033717, A033718, A033712, A033719, A033720, A033721, A033722, A033723, A033724, A033725, A033726, A033727, A033728, A033729, A033730, A033731, A033732, A033733, A033734, A033735, A033736, A033737, A033738, A033739, A033740, A033741, A033742, A033743, A033744, A033745, A033746, A033747, A033748, A033749, A033750, A033751, A033752, A033753, A033754, A033755, A033756, A033757, A033758, A033759, A033760.

Cf. A320305 (diagonal).

Table[Function[k, SeriesCoefficient[Product[EllipticTheta[3, 0, q^d], {d, Divisors[k]}], {q, 0, n}]][i - n + 1], {i, 0, 13}, {n, 0, i}] // Flatten

G.f. of column k: Product_{d|k} theta_3(q^d).

cp's OEIS Frontend

A115979 Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Scheme

Formula

A228447 Expansion of q * (psi(q^3) * psi(q^6)) / (psi(q) * phi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A129576 Expansion of phi(x) * chi(x) * psi(-x^3) in powers of x where phi(), chi(), psi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A239074 Given a circle of radius R into which small circles of radius R/2^n are packed in a "hexagonal pattern" (see Comments), a(n) is the number of points at which a small circle is tangent to the big circle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A320239 Expansion of theta_3(q) * theta_3(q^3) * theta_3(q^5), where theta_3() is the Jacobi theta function.

Views

Author

Keywords

Comments

Links

Crossrefs

A320240 Expansion of theta_3(q) * theta_3(q^3) * theta_3(q^5) * theta_3(q^7), where theta_3() is the Jacobi theta function.

Views

Author

Keywords

Comments

Links

Crossrefs

A374295 a(n) is the smallest positive integer k such that A096936(k) = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A028613 Expansion of theta_3(q) * theta_3(q^12) + theta_2(q) * theta_2(q^12) in powers of q^(1/4).

Views

Author

Keywords

Examples