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A301289
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Coordination sequence for a tetravalent node of type 3.4.3.12 in "cph" 2-D tiling (or net).
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38
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1, 4, 5, 6, 12, 14, 15, 18, 21, 26, 28, 26, 31, 38, 37, 38, 44, 46, 47, 50, 53, 58, 60, 58, 63, 70, 69, 70, 76, 78, 79, 82, 85, 90, 92, 90, 95, 102, 101, 102, 108, 110, 111, 114, 117, 122, 124, 122, 127, 134, 133, 134, 140, 142, 143, 146, 149, 154, 156, 154
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OFFSET
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0,2
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COMMENTS
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Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023
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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 66, bottom row, first tiling.
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LINKS
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FORMULA
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Theorem: G.f. = (1+2*x+4*x^3+3*x^4+6*x^6-4*x^7+6*x^8-2*x^9) / ((1-x)^2*(1+x^2)*(1+x^2+x^4)).
The proof uses the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the first scan. (Not yet added.) The trunks are blue, the branches are red, and the twigs are green. There is mirror symmetry about the Y-axis, and quadrants I and II are essentially identical, as are quadrants III and IV. The counts of the various classes of nodes are given in the second scan, and the corresponding generating functions are in the third scan. Adding up the different terms gives the g.f. stated above. - N. J. A. Sloane, Apr 07 2018
E.g.f.: (6*(3 + (4*exp(x) - 3)*x + 3*sin(x)) - 9*cos(sqrt(3)*x/2)*cosh(x/2) + sqrt(3)*sin(sqrt(3)*x/2)*(8*cosh(x/2) - 5*sinh(x/2)))/9. - Stefano Spezia, Jun 08 2024
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MATHEMATICA
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Join[{1, 4}, LinearRecurrence[{2, -3, 4, -4, 4, -3, 2, -1}, {5, 6, 12, 14, 15, 18, 21, 26}, 100]] (* Jean-François Alcover, Aug 05 2018 *)
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PROG
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(PARI) \\ See Links section.
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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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EXTENSIONS
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STATUS
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approved
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