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A301694
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Expansion of (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)*(1 - x^3)).
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38
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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
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OFFSET
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0,2
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COMMENTS
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Appears to be coordination sequence for node of type V1 in "krd" 2-D tiling (or net). This should be easy to prove by the coloring book method (see link).
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(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
(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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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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