A321015 - OEIS

A321015

Number of isohedral Voronoi parallelotopes in R^n.

1, 2, 2, 4, 3, 6, 4, 7, 4, 6, 3, 10, 3, 7, 6, 9, 3, 10, 3, 10, 7, 6, 3, 15, 5, 6, 6, 11, 3, 14, 3, 11, 6, 6, 8, 16, 3, 6, 6, 15, 3, 15, 3, 10, 10, 6, 3, 19, 6, 10, 6, 10, 3, 14, 7, 16, 6, 6, 3, 22, 3, 6, 11, 13, 7, 14, 3, 10, 6, 15, 3, 23, 3, 6, 10, 10, 8, 14, 3, 19, 8, 6, 3, 23, 7

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OFFSET

1,2

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Marjorie Senechal, Introduction to lattice geometry. In M. Waldschmidt et al., eds., From Number Theory to Physics, pp. 476-495. Springer, Berlin, Heidelberg, 1992. See Cor. 3.7.

FORMULA

a(n) = d(n) + A321013(n) + A321014(n), where d(n) = A000005(n) is the number of divisors of n.

Sum_{k=1..n} a(k) ~ n * (2*log(n) + 4*gamma - 571/168), where gamma is Euler's constant (A001620). - Amiram Eldar, Feb 02 2025

EXAMPLE

Of the five different Voronoi cells of 3-dimensional lattices, only two are isohedral, so a(3) = 2: the cube and the rhombic dodecahedron, the Voronoi cells of the primitive cubic and the face-centered cubic lattices.

MAPLE

d2:=proc(n) local c; if n <= 3 then return(0); fi;

c:=NumberTheory[tau](n)-1;

if (n mod 2)=0 then c:=c-1; fi;

if (n mod 3)=0 then c:=c-1; fi; c; end; # A321014

d3:=proc(n) local c; c:=0;

if (n mod 6)=0 then c:=c+1; fi;

if (n mod 7)=0 then c:=c+1; fi;

if (n mod 8)=0 then c:=c+1; fi; c; end; # A321013

[seq(NumberTheory[tau](n)+d2(n)+d3(n), n=1..120)];

PROG

(PARI) a(n) = 2*numdiv(n) + sum(k = 6, 8, !(n % k)) + n%2 + (n%3>0) - 3; \\ Amiram Eldar, Feb 02 2025