A307204 Coordination sequence for trivalent node of type alpha'''' in the first Moore pentagonal tiling.

1, 3, 6, 6, 12, 18, 24, 30, 30, 33, 48, 57, 60, 60, 72, 84, 78, 81, 96, 111, 96, 102, 120, 138, 114, 123, 144, 165, 132, 144, 168, 192, 150, 165, 192, 219, 168, 186, 216, 246, 186, 207, 240, 273, 204, 228, 264, 300, 222, 249, 288, 327, 240, 270, 312, 354, 258

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 A307204

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) + [18,21,24,27] 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 + 3*x + 6*x^2 + 6*x^3 + 10*x^4 + 12*x^5 + 12*x^6 + 18*x^7 + 7*x^8 + 6*x^10 + 3*x^11 + 12*x^12 + 12*x^13 - 12*x^16 - 6*x^17) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).

a(n) = 2*a(n-4) - a(n-8) for n>17.

(End)

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A310131 A066393 A127777 * A208445 A208796 A319867

Adjacent sequences: A307201 A307202 A307203 * A307205 A307206 A307207

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