%I #16 Sep 08 2022 08:46:14
%S 1,3,6,10,15,22,31,44,62,87,122,171,240,336,471,660,925,1296,1816,
%T 2545,3566,4997,7002,9812,13749,19266,26997,37830,53010,74281,104088,
%U 145855,204382,286394,401315,562350,788003,1104204,1547286,2168163,3038178,4257303,5965624,8359440,11713819,16414204,23000705,32230160
%N Coordination sequence for (3,3,4) tiling of hyperbolic plane.
%H G. C. Greubel, <a href="/A265071/b265071.txt">Table of n, a(n) for n = 0..1000</a>
%H J. W. Cannon, P. Wagreich, <a href="http://dx.doi.org/10.1007/BF01444714">Growth functions of surface groups</a>, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,0,-1).
%F G.f.: (x^3+x^2+x+1)*(x^2+x+1)*(x+1)/(x^6-x^4-x^3-x^2+1).
%F a(n) = a(n-2)+a(n-3)+a(n-4)-a(n-6) for n>6. - _Vincenzo Librandi_, Dec 30 2015
%t CoefficientList[Series[(x^3 + x^2 + x + 1) (x^2 + x + 1) (x + 1)/(x^6 - x^4 - x^3 - x^2 + 1), {x, 0, 60}], x] (* _Vincenzo Librandi_, Dec 30 2015 *)
%o (Magma) I:=[1,3,6,10,15,22,31]; [n le 7 select I[n] else Self(n-2)+Self(n-3)+Self(n-4)- Self(n-6): n in [1..50]]; // _Vincenzo Librandi_, Dec 30 2015
%o (PARI) x='x+O('x^50); Vec((x^3+x^2+x+1)*(x^2+x+1)*(x+1)/(x^6-x^4-x^3-x^2+1)) \\ _G. C. Greubel_, Aug 07 2017
%Y 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.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 29 2015