A301708 Coordination sequence for node of type V1 in "krc" 2-D tiling (or net).
1, 6, 11, 16, 22, 28, 33, 38, 44, 50, 55, 60, 66, 72, 77, 82, 88, 94, 99, 104, 110, 116, 121, 126, 132, 138, 143, 148, 154, 160, 165, 170, 176, 182, 187, 192, 198, 204, 209, 214, 220, 226, 231, 236, 242, 248, 253, 258, 264, 270, 275, 280, 286, 292, 297, 302
Offset: 0
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, 1st tiling.
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..1000 (first 100 terms from Davide M. Proserpio)
- Brian Galebach, Collection of n-Uniform Tilings. See Number 15 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 krc 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 A301708.
- Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Crossrefs
Cf. A301710.
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
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Mathematica
LinearRecurrence[{2,-2,2,-1},{1,6,11,16,22},100] (* Paolo Xausa, Nov 14 2023 *) -
PARI
\\ See Links section.
Formula
G.f. = (x^4+4*x^3+x^2+4*x+1)/((x^2+1)*(x-1)^2); for n>0, a(2*t)=11*t, a(4*t+1)=22*t+6, a(4*t+3)=22*t+16. These should be easy to prove by the coloring book method (see link).
Conjecture: a(n) = (i*((-i)^n - i^n) + 22*n) / 4 where i=sqrt(-1). - Colin Barker, Apr 07 2018
E.g.f.: (2 + 11*exp(x)*x + sin(x))/2. - Stefano Spezia, Jun 08 2024
Extensions
More terms from Davide M. Proserpio, Mar 28 2018
Comments