A307205 Coordination sequence for tetravalent node in the first Moore pentagonal tiling.

1, 4, 8, 14, 19, 24, 29, 36, 44, 48, 54, 58, 69, 68, 77, 80, 94, 88, 100, 102, 119, 108, 123, 124, 144, 128, 146, 146, 169, 148, 169, 168, 194, 168, 192, 190, 219, 188, 215, 212, 244, 208, 238, 234, 269, 228, 261, 256, 294, 248, 284, 278, 319, 268, 307, 300

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 A307205

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

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Formula

For n >= 7, a(n+4) = a(n) + [25,20,23,22] 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 + 4*x + 8*x^2 + 14*x^3 + 17*x^4 + 16*x^5 + 13*x^6 + 8*x^7 + 7*x^8 + 4*x^9 + 4*x^10 - 4*x^13 - 2*x^14) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).

a(n) = 2*a(n-4) - a(n-8) for n>14. (End)

Mathematica

LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {1, 4, 8, 14, 19, 24, 29, 36, 44, 48, 54, 58, 69, 68, 77}, 100] (* Paolo Xausa, Apr 09 2026 *)

Prog

(PARI) \\ See Links section.

Crossrefs

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

Sequence in context: A312622 A312623 A312624 * A312625 A312626 A312627

Adjacent sequences: A307202 A307203 A307204 * A307206 A307207 A307208

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 30 2019

Extensions

Terms a(7)-a(20) (and a corrected a(6)) from Davide M. Proserpio using ToposPro, Apr 01 2019

More terms from Rémy Sigrist, Apr 02 2019

Status

approved