bibkey: barker2018a299289
authors: N. J. A. Sloane, Colin Barker
year: 2018
title: "OEIS A299289, coordination sequence for the tsi 3D uniform tiling"
doi: null
url: https://oeis.org/A299289
claim: "Conjectures from Colin Barker: a(n) = (13*n^2 + 4)/2 for n>0 and even, a(n) = (13*n^2 + 3)/2 for n odd, with the corresponding generating function and recurrence."
strata_touched:
  - D5/S3/StatisticalMechanics/TsiCoordinationSequence
license: citation-only
triage: anchor

OEIS A299289

Verified locator

Source: https://oeis.org/A299289

The entry was read through the plain-text endpoint https://oeis.org/search?q=id:A299289&fmt=text. It was authored by N. J. A. Sloane (February 10, 2018); its first 20 terms were computed by D. M. Proserpio with ToposPro, and it links the Reticular Chemistry Structure Resource page of the tsi net.

Statement

The entry is the coordination sequence of the tsi uniform tiling of 3-space: $a(n)$ counts the vertices at graph distance exactly $n$ from a fixed vertex of its edge graph, with terms $1,8,28,60,106,164,236,\dots$. Its formulas are labelled conjectural:

Conjectures from Colin Barker, Feb 11 2018: (Start) G.f.: (1 + 6x + 12x^2 + 6x^3 + x^4) / ((1 - x)^3(1 + x)). a(n) = (13n^2 + 4) / 2 for n>0 and even. a(n) = (13n^2 + 3) / 2 for n odd. a(n) = 2a(n-1) - 2a(n-3) + a(n-4) for n>4. (End)

Encoding and boundary

The repository presents the net on ${\mathbb{Z}}^3$: square grids in the layers, and between consecutive layers prism slabs whose axes alternate between the two grid directions (each vertex joined to two vertices of the next layer), realized by regular triangular prisms with unit edges. It proves the two closed forms for every $n\ge 1$ from the vertex $(0,0,0)$, through an exact distance formula (layer distance plus the $L^1$ distance to a rectangle determined by the slab orientations). The generating function and the recurrence follow from the closed forms but are not stated separately; the companion partial sums and the other uniform tilings are outside this conclusion.