A307203 Coordination sequence for trivalent node of type alpha'' in the first Moore pentagonal tiling.

1, 3, 7, 10, 14, 21, 26, 32, 38, 46, 51, 56, 61, 71, 73, 78, 84, 94, 95, 100, 107, 117, 117, 122, 130, 140, 139, 144, 153, 163, 161, 166, 176, 186, 183, 188, 199, 209, 205, 210, 222, 232, 227, 232, 245, 255, 249, 254, 268, 278, 271, 276, 291, 301, 293, 298

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

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 A307203

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 >= 10, a(n+4) = a(n) + [23,23,22,22] according as n == [0,1,2,3] mod 4. - Chaim Goodman-Strauss, Mar 31 2019

Conjectures from Colin Barker, Apr 03 2019: (Start)

G.f.: (1 + 2*x + 5*x^2 + 5*x^3 + 7*x^4 + 8*x^5 + 4*x^6 + 8*x^7 + 3*x^8 + 4*x^9 + 2*x^10 - x^12 + x^13 - 4*x^14 + 2*x^15 - 2*x^16) / ((1 - x)^2*(1 + x)*(1 + x^2)^2).

a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + a(n-6) - a(n-7) for n>16.

(End)

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A293788 A319480 A310190 * A347638 A224880 A043722

Adjacent sequences: A307200 A307201 A307202 * A307204 A307205 A307206

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