A298024 Expansion of (x^4+3*x^3+6*x^2+3*x+1)/((1-x)*(1-x^3)).

1, 4, 10, 14, 18, 24, 28, 32, 38, 42, 46, 52, 56, 60, 66, 70, 74, 80, 84, 88, 94, 98, 102, 108, 112, 116, 122, 126, 130, 136, 140, 144, 150, 154, 158, 164, 168, 172, 178, 182, 186, 192, 196, 200, 206, 210, 214, 220, 224, 228, 234, 238, 242, 248, 252, 256, 262

Offset: 0

Author

N. J. A. Sloane, Jan 21 2018

Keywords

• nonn
• easy

Comments

Coordination sequence for Dual(3^3.4^2) tiling with respect to a tetravalent node. This tiling is also called the prismatic pentagonal tiling, or the cem-d net. It is one of the 11 Laves tilings. (The identification of this coordination sequence with the g.f. in the definition was first conjectured by Colin Barker, Jan 22 2018.)

Also, coordination sequence for a tetravalent node in the "krl" 2-D tiling (or net).

Both of these identifications are easily established using the "coloring book" method - see the Goodman-Strauss & Sloane link.

For n>0, this is twice A047386 (numbers congruent to 0 or +-2 mod 7).

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 66, 3rd row, second tiling. (For the krl tiling.)
• B. Gruenbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. See p. 96. (For the Dual(3^3.4^2) tiling.)

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..1000
• Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. Isoperimetric Pentagonal Tilings, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (right).
• Brian Galebach, Collection of n-Uniform Tilings. See Number 4 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 preprint, arXiv:1803.08530 [math.CO], 2018-2019.
• Tom Karzes, Tiling Coordination Sequences
• Frank Morgan, Optimal Pentagonal Tilings, Video, May 2021. [Mentions this tiling]
• Reticular Chemistry Structure Resource (RCSR), The cem-d tiling (or net)
• Reticular Chemistry Structure Resource (RCSR), The krl 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 initial terms
• Rémy Sigrist, PARI program for A298024
• N. J. A. Sloane, Illustration of initial terms [1 (black), 4 (black), 10 (black), 14 (red)]
• N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A301298.

See A298025 for partial sums, A298022 for a trivalent node.

See also A047486.

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

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

CoefficientList[Series[(x^4+3x^3+6x^2+3x+1)/((1-x)(1-x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{1,0,1,-1},{1,4,10,14,18},80] (* Harvey P. Dale, Oct 03 2018 *)

PARI

See Links section.

Formula

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. (Conjectured, correctly, by Colin Barker, Jan 22 2018.)

Extensions

More terms from Rémy Sigrist, Jan 21 2018

Entry revised by N. J. A. Sloane, Mar 25 2018