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A166468
Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
2
1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708582, 2125728, 6377136, 19131264, 57393360, 172178784, 516532464, 1549585728, 4648722192, 13946061600, 41837869872, 125512664832, 376535160174, 1129596977628
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003946, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
LINKS
Index entries for linear recurrences with constant coefficients, signature (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -3).
FORMULA
G.f.: (t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(3*t^12 - 2*t^11 - 2*t^10 - 2*t^9 - 2*t^8 - 2*t^7 - 2*t^6 - 2*t^5 - 2*t^4 - 2*t^3 - 2*t^2 - 2*t + 1).
G.f.: (1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13). - G. C. Greubel, Apr 26 2019
a(n) = -3*a(n-12) + 2*Sum_{k=1..11} a(n-k). - Wesley Ivan Hurt, May 06 2021
MATHEMATICA
CoefficientList[Series[(1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13), {x, 0, 30}], x ] (* Vincenzo Librandi, Apr 29 2014 *)(* modified by G. C. Greubel, Apr 26 2019 *)
coxG[{12, 3, -2, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 09 2018 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)) \\ G. C. Greubel, Apr 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13) )); // G. C. Greubel, Apr 26 2019
(Sage) ((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019
CROSSREFS
Sequence in context: A165184 A165756 A166328 * A166858 A167105 A167649
KEYWORD
nonn,easy
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved