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A301287
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Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).
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38
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1, 3, 6, 7, 8, 15, 18, 17, 20, 25, 28, 29, 30, 35, 40, 39, 40, 47, 50, 49, 52, 57, 60, 61, 62, 67, 72, 71, 72, 79, 82, 81, 84, 89, 92, 93, 94, 99, 104, 103, 104, 111, 114, 113, 116, 121, 124, 125, 126, 131, 136, 135, 136, 143, 146, 145, 148, 153, 156, 157, 158
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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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G.f. = -(2*x^8-2*x^7-x^6-4*x^5-2*x^4-2*x^3-4*x^2-2*x-1) / ((x^2+1)*(x^2+x+1)*(x-1)^2). N. J. A. Sloane, Mar 28 2018 (This is now a theorem. - N. J. A. Sloane, Apr 05 2018)
Theorem: G.f. = (1+2*x+4*x^2+2*x^3+2*x^4+4*x^5+1*x^6+2*x^7-2*x^8) / ((1-x)*(1+x^2)*(1-x^3)).
Proof. This follows by applying the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the links, and the details of the proof (by calculating the generating function) are on the next two scanned pages. - N. J. A. Sloane, Apr 05 2018
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MATHEMATICA
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Join[{1, 3, 6}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {7, 8, 15, 18, 17, 20}, 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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