A307205 - OEIS

%I #22 Apr 03 2019 08:31:09

%S 1,4,8,14,19,24,29,36,44,48,54,58,69,68,77,80,94,88,100,102,119,108,

%T 123,124,144,128,146,146,169,148,169,168,194,168,192,190,219,188,215,

%U 212,244,208,238,234,269,228,261,256,294,248,284,278,319,268,307,300

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

%C 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).

%C The group is p3m1. - _Davide M. Proserpio_, Apr 01 2019

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

%H Rémy Sigrist, <a href="/A307205/b307205.txt">Table of n, a(n) for n = 0..1000</a>

%H Davide M. Proserpio, <a href="/A307201/a307201_1.png">Another drawing of the first Moore tiling</a> {Labels: V1 = alpha'''', V2 = alpha''', V3 = alpha'', V4 = beta, V5 = alpha', V6 = alpha]

%H Rémy Sigrist, <a href="/A307205/a307205.png">Illustration of first terms</a>

%H Rémy Sigrist, <a href="/A307205/a307205.gp.txt">PARI program for A307205</a>

%H N. J. A. Sloane, <a href="/A307201/a307201.png">The first Moore tiling</a> [Constructed by copy-and-paste from the illustration in the patent]

%H N. J. A. Sloane, <a href="/A307201/a307201_2.png">Fundamental cell</a>

%F 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

%F Conjectures from _Colin Barker_, Apr 03 2019: (Start)

%F 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).

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

%F (End)

%o (PARI) See Links section.

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

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Mar 30 2019

%E Terms a(7)-a(20) (and a corrected a(6)) from _Davide M. Proserpio_ using ToposPro, Apr 01 2019

%E More terms from _Rémy Sigrist_, Apr 02 2019