A301298 Expansion of (1 + 4*x + 4*x^2 + 4*x^3 + x^4)/((1 - x)*(1 - x^3)).

1, 5, 9, 14, 19, 23, 28, 33, 37, 42, 47, 51, 56, 61, 65, 70, 75, 79, 84, 89, 93, 98, 103, 107, 112, 117, 121, 126, 131, 135, 140, 145, 149, 154, 159, 163, 168, 173, 177, 182, 187, 191, 196, 201, 205, 210, 215, 219, 224, 229, 233, 238, 243, 247, 252, 257, 261

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.

Sequence in context: A314887 A316324 A247972 * A314888 A314889 A314890

Adjacent sequences: A301295 A301296 A301297 * A301299 A301300 A301301

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 24 2018

Status

approved