A307201 Coordination sequence for trivalent node of type alpha in the first Moore pentagonal tiling.

1, 3, 9, 12, 21, 30, 30, 33, 42, 57, 51, 54, 63, 84, 72, 75, 84, 111, 93, 96, 105, 138, 114, 117, 126, 165, 135, 138, 147, 192, 156, 159, 168, 219, 177, 180, 189, 246, 198, 201, 210, 273, 219, 222, 231, 300, 240, 243, 252, 327, 261, 264, 273, 354, 282, 285

Offset

0,2

Comments

There are six orbits on nodes, and six distinct coordination sequences, which are A307201 (nodes of type alpha), A307202 (alpha'), A307203 (alpha''), A307270 (alpha'''), A307204 (alpha''''), and A307206 (beta).

The group is p3m1. - Davide M. Proserpio, Apr 01 2019

References

Herbert C. Moore, U.S. Patent 928,320, Patented July 20 1909.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..1000

Chaim Goodman-Strauss, Analysis of coordination sequences for this tiling using the Coloring Book method

Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.

Davide M. Proserpio, Another drawing of the first Moore tiling {Labels: V1 = alpha'''', V2 = alpha''', V3 = alpha'', V4 = beta, V5 = alpha', V6 = alpha]

Rémy Sigrist, Illustration of first terms

Rémy Sigrist, PARI program for A307201

N. J. A. Sloane, Chaim Goodman-Strauss, and others, Discussion and analysis of the coordination sequences for this tiling [On the Tiling List archives]

N. J. A. Sloane, The first Moore tiling [Constructed by copy-and-paste from the illustration in the patent]

N. J. A. Sloane, Fundamental cell

Formula

For n >= 1, a(n+4) = a(n) + [21,27,21,21] according as n == [0,1,2,3] mod 4. - Chaim Goodman-Strauss, Mar 31 2019

From Colin Barker, Apr 03 2019: (Start)

G.f.: (1 + 3*x + 9*x^2 + 12*x^3 + 19*x^4 + 24*x^5 + 12*x^6 + 9*x^7 + x^8) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).

a(n) = 2*a(n-4) - a(n-8) for n>8. (End)

E.g.f.: (4 + 3*(1 + x)*cos(x) + 3*(8*x - 1)*cosh(x) + 3*(7*x - 5)*sinh(x))/4. - Stefano Spezia, Sep 07 2022

Prog

(PARI) See Links section.

Crossrefs

Cf. A307201-A307205, A307270, A307271-A307276.

Sequence in context: A155504 A270672 A211217 * A022379 A344720 A303192

Adjacent sequences: A307198 A307199 A307200 * A307202 A307203 A307204

Keyword

nonn

Author

N. J. A. Sloane, Mar 30 2019

Extensions

Terms a(7)-a(20) from Davide M. Proserpio using ToposPro, Apr 01 2019

More terms from Rémy Sigrist, Apr 02 2019

Status

approved