A301716 Coordination sequence for node of type V1 in "kre" 2-D tiling (or net).

1, 6, 12, 18, 18, 30, 36, 36, 48, 48, 54, 66, 66, 72, 78, 84, 90, 96, 102, 102, 114, 120, 120, 132, 132, 138, 150, 150, 156, 162, 168, 174, 180, 186, 186, 198, 204, 204, 216, 216, 222, 234, 234, 240, 246, 252, 258, 264, 270, 270, 282, 288, 288, 300, 300, 306

Offset: 0

Author

N. J. A. Sloane, Mar 26 2018

Keywords

• nonn
• easy

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, 1st row, 3rd tiling.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..1000
• Brian Galebach, Collection of n-Uniform Tilings. See Number 18 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 kre 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.
• Rémy Sigrist, Illustration of first terms
• Rémy Sigrist, PARI program for A301716
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,0,-1).

Crossrefs

Cf. A301718.

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.

Programs

Mathematica

LinearRecurrence[{0,0,1,0,1,0,0,-1},{1,6,12,18,18,30,36,36,48},100] (* Paolo Xausa, Nov 16 2023 *)

PARI

See Links section.

Formula

G.f.: (x^8+6*x^7+12*x^6+17*x^5+12*x^4+17*x^3+12*x^2+6*x+1) / ((1-x^3)*(1-x^5)). - N. J. A. Sloane, Mar 28 2018

Extensions

More terms from Rémy Sigrist, Mar 28 2018