A033687 -id:A033687 - cp's OEIS frontend

A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

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

Offset: 0

Michael Somos, Aug 04 2015

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).

			G.f. = x + x^2 - 2*x^3 - x^4 - 2*x^6 + 2*x^7 + x^8 - 2*x^9 + 2*x^12 + ...

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. A033687, A033762, A097195, A093829, A112848, A113447, A123530, A123863, A227696, A246838, A248897, A253243, A260941, A260942, A260943, A260944.

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

A := Basis( ModularForms( Gamma1(36), 1), 80); A[2] + A[3] - 2*A[4] - A[5] - 2*A[7] + 2*A[8] + A[9] - 2*A[10] + 2*A[13] + 2*A[14] + 2*A[15] - A[17] - 2*A[19] - 4*A[20];

a[ n_] := If[ n < 1, 0, Sum[ {1, 1, 0, -1, -1, 0}[[Mod[ d, 6, 1]]] {1, 0, -2, 0, 1, 0}[[Mod[ n/d, 6, 1]]], {d, Divisors @ n}]]
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -(-1)^#2, # == 3, -2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] EllipticTheta[ 3, 0, q] / (2 q^(1/4) QPochhammer[ q^6]), {q, 0, n}];

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

{a(n) = if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, -2, p%6==5, 1-e%2, e+1)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

Expansion of (a(q) + a(q^2) - 3*a(q^3) - 2*a(q^4) - 3*a(q^6) + 6*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q * phi(q) * psi(-q) * psi(-q^9) / f(-q^6) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^4 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, -3, 1, -2, 1, -2, 1, -2, 0, -3, 1, -1, 1, -3, 1, -2, 1, -2, 1, -2, 1, -3, 1, -1, 1, -3, 0, -2, 1, -2, 1, -2, 1, -3, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, -3, -2, -1, 0, 1, 2, 0, 0, -1, 6, 1, 0, 3, -2, -1, 0, 1, 2, -3, 0, -1, -6, 1, 0, 0, -2, -1, 0, 1, 2, 3, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = -2, if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123863.
a(2*n) = A112848(n). a(2*n + 1) = A123530(n). a(3*n) = -2 * A113447(n). a(3*n + 1) = A227696(n).
a(4*n) = - A112848(n). a(4*n + 1) = A253243(n). a(4*n + 2) = A123530(n). a(4*n + 3) = -2 * A246838(n).
a(6*n) = -2 * A093829(n). a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n). a(6*n + 3) = -2 * A033762(n). a(6*n + 5) = 0.
a(8*n + 1) = A260941(n). a(8*n + 2) = A253243(n). a(8*n + 3) = -2 * A260943(n). a(8*n + 4) = - A123530(n). a(8*n + 5) = 2 * A260942(n). a(8*n + 6) = -2 * A246838(n). a(8*n + 7) = 2 * A260944(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A260958 Expansion of (a(q) - 3a(q^2) + 3a(q^3) - 4a(q^4) + 3a(q^6)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

0, 1, -3, 4, -3, 0, 0, 2, -3, 4, 0, 0, 0, 2, -6, 0, -3, 0, 0, 2, 0, 8, 0, 0, 0, 1, -6, 4, -6, 0, 0, 2, -3, 0, 0, 0, 0, 2, -6, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, -3, 0, -6, 0, 0, 0, -6, 8, 0, 0, 0, 2, -6, 8, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, -6, 4, -6, 0, 0, 2, 0, 4

Offset: 0

Michael Somos, Aug 05 2015

sign

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

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

			G.f. = x - 3*x^2 + 4*x^3 - 3*x^4 + 2*x^7 - 3*x^8 + 4*x^9 + 2*x^13 + ...

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

Cf. A033687, A033762, A097195, A113448, A122861.

a[ n_] := If[ n < 1, 0, {1, -3, 4, -3, 1, 0}[[Mod[ n, 6, 1]]] Sum[ {1, 0, 0, 0, -1, 0}[[Mod[ d, 6, 1]]], {d, Divisors @ n}]];
a[ n_] := If[ n < 1 || Mod[n, 6] == 0, 0, Times @@ (Which[ # == 1, 1, # == 2, -2 - Mod[#2, 2], # == 3, 4, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 3, 0, x^3] QPochhammer[ -x^3, x^6]^2 EllipticTheta[ 2, 0, x^(9/2)] / (2 x^(1/8) QPochhammer[-x, x^2]^2 EllipticTheta[ 3, 0, x^9]), {x, 0, n}];

{a(n) = if( n<1, 0, [0, 1, -3, 4, -3, 1][n%6+1] * sumdiv(n, d, [0, 1, 0, 0, 0, -1][n/d%6+1]))};

{a(n) = my(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)^9 * eta(x^9 + A) * eta(x^36 + A)^2 / (eta(x^2 + A)^4 * eta(x^3 + A)^4 * eta(x^12 + A)^4 * eta(x^18 + A)^3), n))};

Expansion of q * f(q) * phi(q^3) * chi(q^3)^2 * psi(q^9) / (chi(q)^2 * phi(q^9)) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q)^3 * eta(q^4)^2 * eta(q^6)^9 * eta(q^9) * eta(q^36)^2 / (eta(q^2)^4 * eta(q^3)^4 * eta(q^12)^4 * eta(q^18)^3) in powers of q.
Euler transform of period 36 sequence [ -3, 1, 1, -1, -3, -4, -3, -1, 0, 1, -3, -2, -3, 1, 1, -1, -3, -2, -3, -1, 1, 1, -3, -2, -3, 1, 0, -1, -3, -4, -3, -1, 1, 1, -3, -2, ...].
Moebius transform is period 36 sequence [ 1, -4, 3, 0, -1, 0, 1, 0, 0, 4, -1, 0, 1, -4, -3, 0, -1, 0, 1, 0, 3, 4, -1, 0, 1, -4, 0, 0, -1, 0, 1, 0, -3, 4, -1, 0, ...].
a(2*n) = A113448(n). a(3*n + 1) = A122861(n). a(6*n) = 0. a(6*n + 1) = A097195(n). a(6*n + 2) = a(12*n + 4) = -3 * A033687(n). a(6*n + 3) = 4 * A033762(n). a(6*n + 5) = a(12*n + 10) = 0.

A261884 Expansion of (a(q) - a(q^2) - 2a(q^3) + 2a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

Original entry on oeis.org

1, -1, -1, 1, 0, 1, 2, -1, -1, 0, 0, -1, 2, -2, 0, 1, 0, 1, 2, 0, -2, 0, 0, 1, 1, -2, -1, 2, 0, 0, 2, -1, 0, 0, 0, -1, 2, -2, -2, 0, 0, 2, 2, 0, 0, 0, 0, -1, 3, -1, 0, 2, 0, 1, 0, -2, -2, 0, 0, 0, 2, -2, -2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 2, -2, -1, 2, 0, 2, 2, 0

Offset: 1

Michael Somos, Sep 04 2015

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

			G.f. = x - x^2 - x^3 + x^4 + x^6 + 2*x^7 - x^8 - x^9 - x^12 + 2*x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A033687, A033762, A093829, A097195, A035178 (apparently gives the absolute values).

Cf. A004016, A005882, A005928.

A004016[q_] := (QPochhammer[q]^3 + 9*q*QPochhammer[q^9]^3)/ QPochhammer[q^3]; A261884[n_] := SeriesCoefficient[(A004016[q] - A004016[q^2] - 2*A004016[q^3] + 2*A004016[q^6])/6, {q, 0, n}]; Table[A261884[n], {n, 1, 50}] (* G. C. Greubel, Sep 24 2017 *)

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, (-1)^e, p==3, -1, p%6==1, e+1, 1-e%2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) - x * eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^4 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2), n))};

Moebius transform is period 18 sequence [ 1, -2, -2, 2, -1, 4, 1, -2, 0, 2, -1, -4, 1, -2, 2, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} F(x^(6*k - 5)) - F(x^(6*k - 3)) + F(x^(6*k - 1)) where F(x) := x / (1 + x + x^2).
a(n) = A093829(n) unless n == 0 (mod 3). a(2*n) = - a(n). a(3*n + 1) = A033687(n).
a(6*n + 1) = A097195(n). a(6*n + 3) = - A033762(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(18*sqrt(3)) = 0.100766631346... . - Amiram Eldar, Nov 23 2023

A305185 a(n) minimizes the maximum norm of elements in a complete residue system of Eisenstein integers modulo n.

Original entry on oeis.org

0, 1, 3, 4, 7, 12, 13, 19, 27, 28, 37, 48, 49, 61, 75, 76, 91, 108, 109, 127, 147, 148, 169, 192, 193, 217, 243, 244, 271, 300, 301, 331, 363, 364, 397, 432, 433, 469, 507, 508, 547, 588, 589, 631, 675, 676, 721, 768, 769, 817, 867, 868, 919, 972, 973, 1027, 1083, 1084, 1141, 1200

Offset: 1

Jianing Song, May 27 2018

From Jianing Song, May 05 2019: (Start)
For any Eisenstein integer w != 0, let R(w) be any set of N(w) Eisenstein integers such that no two numbers are congruent modulo w, then we intend to find the smallest possible value of max_{s in R(w)} N(s). Here N(w) is the norm of w.
If we can find a set of complex numbers A such that: (i) for any Eisenstein integer x, r in A, |r| <= |r - x|; (ii) every complex number z can be uniquely represented as z = x + r, where x is an Eisenstein integer, r is in A, then S(w) = {r*w : r is in A} is a complete residue system modulo w formed by choosing one element with the minimal norm in each residue class modulo w (there may be more than one element whose norms are minimal in one residue class). As a result, the smallest possible value of max_{s in R(w)} N(s) is max_{s in S(w)} N(s). For more details, see my further notes in the Link section.
Now, for positive integers n, we find the value of max_{s in S(n)} N(s) over the ring of Eisenstein integers. Let A be the set shown in Page 5, Figure 2 in my further notes on this sequence (see Links section below), and S(w) = {r*w : r in A}. For n >= 2, note that for any s in S(n), s != 0, there exists some s' in S(n) such that s/s' is an Eisenstein unit and arg(s') is in the range [-Pi/6, Pi/6]. Let s' = (x + y*sqrt(3)*i)/2 where x and y have the same parity, 0 < x <= n and -x/3 <= y <= x/3, then N(s) = N(s') = (x^2 + 3*y^2)/4. For fixed x >= 2, we have max |y| = x - 2*ceiling(x/3) so max N(s') = max_{x=2..n} (x^2 + 3*(x - 2*ceiling(x/3))^2)/4 = (n^2 + 3*(n - 2*ceiling(n/3))^2)/4. (End)

			In the following examples let w = (-1 + sqrt(-3))/2. Let A be the set shown in Page 5, Figure 2 in my further notes on this sequence, and S(w) = {r*w : r is in A}.
n = 1: S(1) = {0}, so a(1) = max_{s in S(1)} N(s) = 0.
n = 2: S(2) = {0, 1, w, w+1}, so a(2) = max_{s in S(2)} N(s) = 1.
n = 3: S(3) = {0, 1, -1, w, w+1, -w, -w-1, w+2, -w-2}, so a(3) = max_{s in S(3)} N(s) = 3.

Jianing Song, Table of n, a(n) for n = 1..10000
Jianing Song, Further notes on A305185

Cf. A004016, A033687.

a(n) = if(n>1, n^2 - 3*n*ceil(n/3) + 3*ceil(n/3)^2, 0) \\ Jianing Song, May 12 2019

From Jianing Song, May 05 2019: (Start)
a(1) = 0; for n >= 2, a(n) = (n^2 + 3*(n - 2*ceiling(n/3))^2)/4 = n^2 - 3*n*ceiling(n/3) + 3*ceiling(n/3)^2.
For k >= 1, a(3*k-1) = 3*k^2 - 3*k + 1, a(3*k) = 3*k^2, a(3*k+1) = 3*k^2 + 1.
G.f.: (x^2*(1 + x^2)*(1 + 2*x - x^3 + x^4))/((1 - x)^3*(1 + x + x^2)^2). (End)

Entry rewritten by Jianing Song, May 05 2019

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A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

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A260958 Expansion of (a(q) - 3*a(q^2) + 3*a(q^3) - 4*a(q^4) + 3*a(q^6)) / 6 in powers of q where a() is a cubic AGM theta function.

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A261884 Expansion of (a(q) - a(q^2) - 2*a(q^3) + 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

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A305185 a(n) minimizes the maximum norm of elements in a complete residue system of Eisenstein integers modulo n.

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A260958 Expansion of (a(q) - 3a(q^2) + 3a(q^3) - 4a(q^4) + 3a(q^6)) / 6 in powers of q where a() is a cubic AGM theta function.

A261884 Expansion of (a(q) - a(q^2) - 2a(q^3) + 2a(q^6)) / 6 in powers of q where a() is a cubic AGM function.