A113063 -id:A113063 - cp's OEIS frontend

A113062 Expansion of theta series of hexagonal net with respect to a node.

1, 3, 0, 6, 3, 0, 0, 6, 0, 6, 0, 0, 6, 6, 0, 0, 3, 0, 0, 6, 0, 12, 0, 0, 0, 3, 0, 6, 6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, 12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, 6, 0, 0, 0, 0, 12, 0, 0, 0, 6, 0, 12, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 6, 6, 0, 0, 6, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 0

Offset: 0

Michael Somos, Oct 13 2005

The hexagonal net is the familiar 2-dimensional honeycomb (not a lattice) in which each node has 3 neighbors.

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

			G.f. = 1 + 3*q + 6*q^3 + 3*q^4 + 6*q^7 + 6*q^9 + 6*q^12 + 6*q^13 + 3*q^16 + ...

A. F. Wells, Structural Inorganic Chemistry, Oxford, 5th ed., 1984; see Fig. 3.9(a.1).

T. D. Noe, Table of n, a(n) for n = 0..1000
George L. Hall, Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From _N. J. A. Sloane_, Dec 18 2012
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A004016, A005882, A005928, A033685, A113063, A275486.

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

a[0] = 1; a[n_] := 3*DivisorSum[n, {0, 1, -1, 1, 1, -1, -1, 1, -1}[[Mod[#, 9]+1]]&]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 04 2015, after 1st PARI script *)

{a(n) = if( n<1, n==0, 3 * sumdiv(n, d, [ 0, 1, -1, 1, 1, -1, -1, 1, -1][d%9+1]))};

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

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

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 6 * x * eta(x^9 + A)^3) / eta(x^3 + A), n))}; /* Michael Somos, Aug 15 2006 */

Moebius transform is period 9 sequence [ 3, -3, 3, 3, -3, -3, 3, -3, 0, ...].
Expansion of a(q^3) + c(q^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Aug 15 2006
For n>0, a(n) = 3*b(n) where b(n)=A113063(n) is multiplicative and b(p^e) = 2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
a(3*n + 2) = 0. a(3*n + 1) = A005882(n) = A033685(3*n + 1) = -A005928(3*n + 1). a(3*n) = A004016(n) = A005928(3*n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Dec 28 2023

Definition corrected Michael Somos, Oct 17 2005

A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 27 2006

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

Denoted by lambda(n) on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - Michael Somos, Dec 10 2017

			G.f. = q - 2*q^3 + q^4 + 2*q^7 - 2*q^9 - 2*q^12 + 2*q^13 + q^16 + 2*q^19 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv preprint arXiv:1610.07793 [math.NT], 2016.

Cf. A002324, A005928, A033687, A113063.

A123477 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if modp(p,6) = 1 then
            a := a*(e+1) ;
        elif modp(p,6) in {2,5} then
            a := a*(1+(-1)^e)/2 ;
        elif e > 0 then
            a := -2*a ;
        end if;
    end do:
    a ;
end proc:
seq(A123477(n),n=1..100) ; # R. J. Mathar, Feb 22 2021

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, -3, 1, -1, 3, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* Michael Somos, Dec 10 2017 *)

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9+1]))};

{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==3, -2, p%6==1, e+1, !(e%2))))};

Moebius transform is period 9 sequence [1, -1, -3, 1, -1, 3, 1, -1, 0, ...].
a(n) is multiplicative and a(p^e) = -2 if p = 3 and e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
a(3*n + 2) = 0. a(3*n + 1) = A033687(n), a(3*n) = -2*A002324(n).
-3*a(n) = A005928(n) unless n=0. |a(n)| = A113063(n).

cp's OEIS Frontend

A113062 Expansion of theta series of hexagonal net with respect to a node.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

Extensions

A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula