A301674 Coordination sequence for node of type V1 in "krs" 2-D tiling (or net).

1, 4, 8, 14, 16, 26, 22, 34, 36, 38, 44, 54, 46, 62, 64, 62, 72, 82, 70, 90, 92, 86, 100, 110, 94, 118, 120, 110, 128, 138, 118, 146, 148, 134, 156, 166, 142, 174, 176, 158, 184, 194, 166, 202, 204, 182, 212, 222, 190, 230, 232, 206, 240, 250, 214, 258, 260

Offset

0,2

Comments

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 67, bottom row, 2nd tiling.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..1000

Brian Galebach, Collection of n-Uniform Tilings. See Number 6 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 krs 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 A301674

Index entries for linear recurrences with constant coefficients, signature (-1,0,2,2,0,-1,-1).

Formula

(a) G.f. = -(2*x^8-x^7-5*x^6-18*x^5-20*x^4-20*x^3-12*x^2-5*x-1)/((x+1)*(x-1)^2*(x^2+x+1)^2). (b) Satisfies the recurrence {( - 2*n^5 - 13*n^4 - 22*n^3 + 7*n^2 + 30*n)*a(n) + ( - 2*n^5 - 13*n^4 - 25*n^3 + n^2 + 39*n)*a(n + 1) + ( - 6*n^2 + 6*n)*a(n + 2) + (2*n^5 + 7*n^4 + 7*n^3 - 7*n^2 - 9*n)*a(n + 3) + (2*n^5 + 7*n^4 + 4*n^3 - 7*n^2 - 6*n)*a(n + 4) = 0, a(0) = 1, a(1) = 4, a(2) = 8, a(3) = 14, a(4) = 16, a(5) = 26}. - N. J. A. Sloane, Mar 28 2018

Equivalent conjecture: 9*a(n) = 40*n -18*(-1)^n -6*(-1)^n*A076118(n+1) +6*A049347(n) -4*A049347(n-1). - R. J. Mathar, Apr 01 2018

Mathematica

LinearRecurrence[{-1, 0, 2, 2, 0, -1, -1}, {1, 4, 8, 14, 16, 26, 22, 34, 36}, 100] (* Paolo Xausa, Nov 15 2023 *)

Prog

(PARI) See Links section.

Crossrefs

Cf. A301676.

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: A312329 A312330 A312331 * A312332 A312333 A312334

Adjacent sequences: A301671 A301672 A301673 * A301675 A301676 A301677

Keyword

nonn

Author

N. J. A. Sloane, Mar 25 2018

Extensions

More terms from Rémy Sigrist, Mar 28 2018

Status

approved