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A163876
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Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
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27
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1, 3, 6, 12, 24, 48, 93, 180, 351, 684, 1332, 2592, 5046, 9825, 19128, 37239, 72498, 141144, 274788, 534972, 1041513, 2027676, 3947595, 7685400, 14962368, 29129580, 56711106, 110408373, 214949232, 418475259, 814711182, 1586125572, 3087958512
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OFFSET
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0,2
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COMMENTS
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Also, coordination sequence for (6,6,6) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The initial terms coincide with those of A003945, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 1, 1, -1).
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FORMULA
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G.f.: (x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/(x^6 - x^5 - x^4 - x^3 - x^2 - x + 1).
G.f.: (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7). - G. C. Greubel, Apr 25 2019
a(n) = -a(n-6) + Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 07 2021
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MATHEMATICA
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coxG[{6, 1, -1, 40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 22 2015 *)
CoefficientList[Series[(1+x)*(1-x^6)/(1-2*x+2*x^6-x^7), {x, 0, 40}], x] (* G. C. Greubel, Aug 06 2017, modified Apr 25 2019 *)
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PROG
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(PARI) x='x+O('x^40); Vec((x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1)/(x^6-x^5- x^4-x^3-x^2-x+1)) \\ G. C. Greubel, Aug 06 2017
(MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7) )); // G. C. Greubel, Apr 25 2019
(Sage) ((1+x)*(1-x^6)/(1-2*x+2*x^6-x^7)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019
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CROSSREFS
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Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Sequence in context: A001668 A080616 A090572 * A033893 A006851 A164352
Adjacent sequences: A163873 A163874 A163875 * A163877 A163878 A163879
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KEYWORD
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nonn,easy
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AUTHOR
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John Cannon and N. J. A. Sloane, Dec 03 2009
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STATUS
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approved
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