A307270 - OEIS

%I #15 Apr 03 2019 08:31:28

%S 1,3,6,9,13,18,27,31,34,41,50,58,61,64,75,81,82,85,100,104,103,106,

%T 125,127,124,127,150,150,145,148,175,173,166,169,200,196,187,190,225,

%U 219,208,211,250,242,229,232,275,265,250,253,300,288,271,274,325,311

%N Coordination sequence for trivalent node of type alpha''' 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="/A307270/b307270.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="/A307270/a307270.png">Illustration of first terms</a>

%H Rémy Sigrist, <a href="/A307270/a307270.gp.txt">PARI program for A307270</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 >= 10, a(n+4) = a(n) + [21,21,25,23] 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 + 3*x + 6*x^2 + 9*x^3 + 11*x^4 + 12*x^5 + 15*x^6 + 13*x^7 + 9*x^8 + 8*x^9 + 2*x^10 + 5*x^11 + 6*x^12 + 2*x^14 - 4*x^15 - 6*x^16 - 2*x^17) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).

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

%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