OFFSET
0,2
COMMENTS
Coordination sequence for pentavalent node in the "krl" 2-D tiling (or net). (This is easily established using the "coloring book" method - see the Goodman-Strauss & Sloane link.)
Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 3rd row, second tiling.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Brian Galebach, Collection of n-Uniform Tilings. See Number 4 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Reticular Chemistry Structure Resource (RCSR), The krl tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
FORMULA
G.f.: (1 + 4*x + 4*x^2 + 4*x^3 + x^4)/((1 - x)*(1 - x^3)).
a(n) = 5*n - floor((n + 1)/3) for n>0, a(0)=1. - Bruno Berselli, Mar 26 2018
MATHEMATICA
CoefficientList[Series[(x^4 + 4 x^3 + 4 x^2 + 4 x + 1) / ((1 -x) (1 - x^3)), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 26 2018 *)
PROG
(Magma) I:=[1, 5, 9, 14, 19]; [n le 5 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Mar 26 2018
(Magma) [n eq 0 select 1 else 5*n-Floor((n+1)/3): n in [0..60]]; // Bruno Berselli, Mar 26 2018
(PARI) lista(nn) = {x='x+O('x^nn); Vec((x^4+4*x^3+4*x^2+4*x+1)/((1-x)*(1-x^3)))} \\ Altug Alkan, Mar 26 2018
CROSSREFS
Cf. A298024.
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 24 2018
STATUS
approved