A301710 Coordination sequence for node of type V2 in "krc" 2-D tiling (or net).

1, 5, 11, 17, 22, 27, 33, 39, 44, 49, 55, 61, 66, 71, 77, 83, 88, 93, 99, 105, 110, 115, 121, 127, 132, 137, 143, 149, 154, 159, 165, 171, 176, 181, 187, 193, 198, 203, 209, 215, 220, 225, 231, 237, 242, 247, 253, 259, 264, 269, 275, 281, 286, 291, 297, 303

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, 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 A301710.

Index entries for linear recurrences with constant coefficients, signature (2, -2, 2, -1).

Formula

G.f.: (x^4+3*x^3+3*x^2+3*x+1)/((x^2+1)*(x-1)^2); for n>0, a(2*t)=11*t, a(4*t+1)=22*t+5, a(4*t+3)=22*t+17. These should be easy to prove by the coloring book method (see link).

a(n) = ((-i)^(1+n) + i^(1+n) + 22*n) / 4 for n>0, where i=sqrt(-1) (conjectured). - Colin Barker, Apr 07 2018

Mathematica

LinearRecurrence[{2, -2, 2, -1}, {1, 5, 11, 17, 22}, 100] (* Paolo Xausa, Nov 14 2023 *)

Prog

(PARI) See Links section.

Crossrefs

Cf. A301708.

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: A314209 A314210 A314211 * A314212 A314213 A314214

Adjacent sequences: A301707 A301708 A301709 * A301711 A301712 A301713

Keyword

nonn

Author

N. J. A. Sloane, Mar 26 2018

Extensions

More terms from Davide M. Proserpio, Mar 28 2018

Status

approved