A301694 Expansion of (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)*(1 - x^3)).
1, 6, 10, 16, 22, 26, 32, 38, 42, 48, 54, 58, 64, 70, 74, 80, 86, 90, 96, 102, 106, 112, 118, 122, 128, 134, 138, 144, 150, 154, 160, 166, 170, 176, 182, 186, 192, 198, 202, 208, 214, 218, 224, 230, 234, 240, 246, 250, 256, 262, 266, 272, 278, 282, 288, 294
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, 2nd tiling.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..5000
- Brian Galebach, Collection of n-Uniform Tilings. See Number 14 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 krd 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.
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Crossrefs
Cf. A219529.
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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Magma
I:=[1,6,10,16,22]; [n le 5 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Mar 26 2018
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Magma
[n eq 0 select 1 else 6*n-2*Floor((n+1)/3): n in [0..60]]; // Bruno Berselli, Mar 26 2018
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Mathematica
CoefficientList[Series[(x^4 + 5 x^3 + 4 x^2 + 5 x + 1) / ((1 - x) (1 - x^3)), {x, 0, 80}], x] (* Vincenzo Librandi, Mar 26 2018 *) -
PARI
lista(nn) = {x='x+O('x^nn); Vec((x^4+5*x^3+4*x^2+5*x+1)/((1-x)*(1-x^3)))} \\ Altug Alkan, Mar 26 2018
Formula
G.f.: (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)*(1 - x^3)).
a(n) = 6*n - 2*floor((n + 1)/3) for n>0, a(0)=1. - Bruno Berselli, Mar 26 2018
Comments