A301712 Coordination sequence for node of type V1 in "usm" 2-D tiling (or net).

1, 5, 10, 16, 22, 27, 33, 38, 43, 49, 53, 59, 65, 70, 77, 81, 86, 92, 96, 103, 108, 113, 120, 124, 130, 135, 139, 146, 151, 157, 163, 167, 173, 178, 183, 189, 194, 200, 206, 211, 216, 221, 226, 232, 238, 243, 249, 254, 259, 265, 269, 275, 281, 286, 293, 297, 302, 308, 312, 319, 324, 329, 336, 340

Offset

0,2

Comments

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023

References

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 2nd row, 2nd tiling.

Links

Ray Chandler, Table of n, a(n) for n = 0..1000

Brian Galebach, Collection of n-Uniform Tilings. See Number 16 from the list of 20 2-uniform tilings.

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

Reticular Chemistry Structure Resource (RCSR), The usm 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 (1,0,0,-1,2,-1,0,0,1,-1).

Formula

G.f.: -(-x^10-4*x^9-5*x^8-6*x^7-7*x^6-8*x^5-7*x^4-6*x^3-5*x^2-4*x-1)/(x^10-x^9+x^6-2*x^5+x^4-x+1). - N. J. A. Sloane, Mar 29 2018

Equivalent conjecture: 5*a(n) = 27*n -b(n) -5*A014017(n-2) for n>0, where b(n) = 2,-1,1,-2,0 (5-periodic) for n>=1. - R. J. Mathar, Mar 30 2018

Mathematica

LinearRecurrence[{1, 0, 0, -1, 2, -1, 0, 0, 1, -1}, {1, 5, 10, 16, 22, 27, 33, 38, 43, 49, 53}, 100] (* Paolo Xausa, Nov 16 2023 *)

Crossrefs

Cf. A301714.

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: A301697 A313916 A313917 * A313918 A313919 A313920

Adjacent sequences: A301709 A301710 A301711 * A301713 A301714 A301715

Keyword

nonn

Author

N. J. A. Sloane, Mar 26 2018

Extensions

a(11)-a(100) from Davide M. Proserpio, Mar 28 2018

Status

approved