A045833 -id:A045833 - cp's OEIS frontend

A202822 Numbers of the form 3*(x^2 + xy + y^2 + x + y) + 1 where x and y are integers.

1, 4, 7, 13, 16, 19, 25, 28, 31, 37, 43, 49, 52, 61, 64, 67, 73, 76, 79, 91, 97, 100, 103, 109, 112, 121, 124, 127, 133, 139, 148, 151, 157, 163, 169, 172, 175, 181, 193, 196, 199, 208, 211, 217, 223, 229, 241, 244, 247, 256, 259, 268, 271, 277, 283, 289, 292

Offset: 1

Michael Somos, Dec 25 2011

nonn

Closed under multiplication.

Löschian numbers of the form 3*k+1. - Altug Alkan, Nov 18 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.

Cf. A033685, A045833.

Subsequence of A003136, A260682 (subsequence).

a202822 n = a202822_list !! (n-1)
a202822_list = filter ((== 1) . flip mod 3) a003136_list
-- Reinhard Zumkeller, Nov 16 2015

nf[{i_,j_}]:=3(i^2+i*j+j^2+i+j)+1; Union[nf/@Tuples[Range[-10,10],2]] (* Harvey P. Dale, Dec 31 2011 *)

isA(n) = if(n%3 == 0, 0, 0 != sumdiv( n, d, kronecker( -3, d)))

x='x+O('x^500); p=eta(x)^3/eta(x^3); for(n=0, 499, if(polcoeff(p, n) != 0 && n%3==1, print1(n, ", "))) \\ Altug Alkan, Nov 18 2015

is(n)=(n%3==1) && #bnfisintnorm(bnfinit(z^2+z+1), n); \\ Joerg Arndt, Jan 04 2016

list(lim)=my(v=List(), y, t); for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, if((x-y)%3, listput(v, t)); y++)); Set(v) \\ Charles R Greathouse IV, Jul 05 2017

A033685(n) != 0 if and only if n is in the set.

A033685 Theta series of hexagonal lattice A_2 with respect to deep hole.

Original entry on oeis.org

0, 3, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 3, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 9, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0

Offset: 0

N. J. A. Sloane

nonn

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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. = 3*x + 3*x^4 + 6*x^7 + 6*x^13 + 3*x^16 + 6*x^19 + 3*x^25 + 6*x^28 + ...
G.f. = 3*q^(1/3) + 3*q^(4/3) + 6*q^(7/3) + 6*q^(13/3) + 3*q^(16/3) + 6*q^(19/3) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004016, A005882, A005928, A033687, A045833, A217219, A248897.

Cf. A000122, A000700, A010054, A121373.

a[n_] := If[Mod[n, 3] != 1, 0, 3*DivisorSum[n, KroneckerSymbol[#, 3]&]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 03 2015, adapted from PARI *)
s = 3q*(QPochhammer[q^9]^3/QPochhammer[q^3])+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)

{a(n) = if( (n<0) || (n%3 != 1), 0, 3 * sumdiv( n, d, kronecker( d, 3)))}; \\ Michael Somos, Jul 16 2005

{a(n) = my(A); if( (n<0) || (n%3 != 1), 0, n = n\3; A = x * O(x^n); 3 * polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ Michael Somos, Jul 16 2005

a(3*n) = a(3*n + 2) = 0.
a(3*n + 1) = A005882(n) = 3 * A033687(n) = -A005928(3*n + 1) = A004016(3*n + 1) / 2.
Expansion of 3 * eta(q^3)^3 / eta(q) in powers of q^(1/3).
G.f.: 3 * x * Product_{k>0} (1 - x^(9*k))^3 / (1 - x^(3*k)) = 3 * Sum_{k>0} x^k * (1 - x^k) * (1 - x^(2*k)) * (1 - x^(4*k)) / (1 - x^(9*k)). - Michael Somos, Jul 15 2005
Expansion of c(x^3) in powers of x where c(x) is a cubic AGM theta function. - Michael Somos, Oct 17 2006
From Michael Somos, Dec 25 2011: (Start)
G.f.: Sum_{i, j in Z} x^(3 * (i^2 + i*j + j^2 + i + j) + 1).
G.f.: Sum_{i, j, k} x^(3 * Q(i, j, k) - 2) where Q(i, j, k) = i*i + j*j + k*k + i*j + i*k + j*k and the sum is over all integer i, j, k where i + j + k = 1. (End)
a(n) = A217219(n)/2. - N. J. A. Sloane, Oct 05 2012
Expansion of 2 * x * psi(x^6) * f(x^6, x^12) + x * phi(x^3) * f(x^3, x^15) in powers of x where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 09 2018
From Amiram Eldar, Oct 13 2022: (Start)
a(n) = 3*A045833(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). (End)

A217219 Theta series of planar hexagonal net (honeycomb) with respect to deep hole.

Original entry on oeis.org

0, 6, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 0, 12, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 12, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 12, 0

Offset: 0

N. J. A. Sloane, Oct 05 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A033685, A045833, A275486.

terms = 105; s = 6 q QPochhammer[q^9]^3/QPochhammer[q^3] + O[q]^(terms+5); CoefficientList[s, q][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017, after Michael Somos *)
CoefficientList[Series[6 q QPochhammer[q^9]^3/QPochhammer[q^3], {q, 0, 100}], q] (* G. C. Greubel, Aug 10 2018 *)

my(q='q+O('q^100)); concat([0], Vec(6*q*eta(q^9)^3/eta(q^3))) \\ G. C. Greubel, Aug 10 2018

See A033685, A045833.
a(n) = 2*A033685(n) = 6*A045833(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Oct 13 2022

Name edited by Andrey Zabolotskiy, Jun 21 2022

A378006 Square table read by descending antidiagonals: the k-th column has Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Jianing Song, Nov 14 2024

For fixed k, we have Product_{chi} L(chi,s) = Product_{p not dividing k} 1/(1 - 1/p^(ord(p,k)*s))^(phi(k)/ord(p,k)), where phi = A000010 is the Euler totient function and ord(a,k) is the multiplicative order of a modulo k; see Section 3.4 of Chapter VI, Proposition 13, page 72 of J.-P. Serre, A Course in Arithmetic. Using the series expansion of 1/(1-x)^r, we get Product_{chi} L(chi,s) = Product_{p not dividing k} (Sum_{n>=0} binomial(n+phi(k)/ord(p,k)-1,phi(k)/ord(p,k)-1)/p^(ord(p,k)*s)), giving us the formula to calculate T(n,k).

From the formula we can wee that T(n,k) = 0 unless n == 1 (mod k). A378007 is the condensed version giving only {T(k*n+1,k)}.

			Table starts
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 2, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 2, 0, 0, 2, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 2, 0, 0, 0, ...
  1, 1, 0, 1, 0, 0, 0, 2, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
See A378007 for more details.

Jianing Song, Table of n, a(n) for n = 1..11325 (the first 150 diagonals, with n+k = 2..151)
J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973.

Columns: A000012 (k=1), A000035 (k=2), A045833 (k=3), A008442 (k=4).

Cf. A378007.

A378006(n,k) = {
my(f = factor(n), res = 1); for(i=1, #f~, if(k % f[i,1] == 0, return(0));
my(d = znorder(Mod(f[i,1],k))); if(f[i,2] % d != 0, return(0), my(m = f[i,2]/d, r = eulerphi(k)/d); res *= binomial(m+r-1,r-1)));
res;}

Each column is multiplicative: T(p^e,k) = 0 if p divides k; 0 if e is not divisible by ord(p,k); binomial(e/ord(p,k)+phi(k)/ord(p,k)-1,phi(k)/ord(p,k)-1) otherwise.

For odd k, T(2*k,n) = T(k,n) for odd n, 0 for even n.

A132978 Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 7, 15, 32, 63, 114, 201, 350, 591, 967, 1554, 2468, 3855, 5916, 8970, 13471, 20007, 29384, 42771, 61784, 88530, 125838, 177642, 249230, 347484, 481506, 663549, 909788, 1241127, 1684824, 2276781, 3063657, 4105275, 5478698, 7283709, 9648360, 12735471

Offset: 0

Michael Somos, Sep 07 2007

nonn

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. = 1 + 3*x + 7*x^2 + 15*x^3 + 32*x^4 + 63*x^5 + 114*x^6 + 201*x^7 + ...
G.f. = q^2 + 3*q^5 + 7*q^8 + 15*q^11 + 32*q^14 + 63*q^17 + 114*q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A045833, A132974, A132975.

a[ n_] := SeriesCoefficient[ 2^(1/2) x^(-5/8) EllipticTheta[ 3, 0, x^3]  QPochhammer[ x, -x] EllipticTheta[ 2, Pi/4, x^(3/2)]^3 / EllipticTheta[ 2, Pi/4, x^(1/2)]^4, {x, 0, n}] // Simplify;
nmax=60; CoefficientList[Series[Product[(1+x^(3*k))^3 * (1-x^(3*k))^4 * (1+x^(6*k)) / ( (1-x^k)^4 * (1+x^k) * (1+x^(2*k))^3),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)

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

Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.
Expansion of psi(-x^3)^3 * f(-x, x^2) / psi(-x)^4 in powers of x where psi(), f(,) are Ramanujan theta functions.
Expansion of q^(-2/3) * (eta(q^2) * eta(q^6))^2 * eta(q^3) * eta(q^12) / ( eta(q)* eta(q^4) )^3 in powers of q.
Euler transform of period 12 sequence [ 3, 1, 2, 4, 3, -2, 3, 4, 2, 1, 3, 0, ...].
a(n) = A132975(3*n + 2).
Convolution of A132974 and A045833.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

cp's OEIS Frontend

A202822 Numbers of the form 3*(x^2 + xy + y^2 + x + y) + 1 where x and y are integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

PARI

Formula

A033685 Theta series of hexagonal lattice A_2 with respect to deep hole.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A217219 Theta series of planar hexagonal net (honeycomb) with respect to deep hole.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A378006 Square table read by descending antidiagonals: the k-th column has Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A132978 Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula