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%I #47 Jun 14 2020 01:58:39
%S 1,1,1,3,4,5,8,12,17,25,37,54,79,116,170,249,365,535,784,1149,1684,
%T 2468,3617,5301,7769,11386,16687,24456,35842,52529,76985,112827,
%U 165356,242341,355168,520524,762865,1118033,1638557,2401422,3519455,5158012,7559434
%N a(1)=a(2)=a(3)=1, a(4)=3; thereafter a(n) = a(n-1) + a(n-3).
%C Also (essentially), coordination sequence for (2,4,infinity) tiling of hyperbolic plane. - _N. J. A. Sloane_, Dec 29 2015
%C Column sums of shifted (1,2) Pascal array:
%C 1 1 1 1 1 1 1 1 1
%C ......2 3 4 5 6 7
%C ............2 5 9
%C .................
%C ----------------- +
%C 1 1 1 3 4 5 8 ...
%C a(n+1) is the number of multus bitstrings of length n with no runs of 2 0's. - _Steven Finch_, Mar 25 2020
%C From _Areebah Mahdia_ and _Greg Dresden_, Jun 13 2020: (Start)
%C For n >= 5, a(n) gives the number of ways to tile the following board of length n-3 with squares and trominos:
%C ._ _
%C |_|_|
%C |_|_|_ _ _ _ _
%C |_|_|_|_|_|_|_| ... . (End)
%H Reinhard Zumkeller, <a href="/A179070/b179070.txt">Table of n, a(n) for n = 1..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 Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus distributions</a>, arXiv:2003.09458 [math.CO], 2020.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1).
%F a(n) = A000930(n-1) + A000930(n-4).
%F G.f.: x - x^2*(1+2*x^2) / ( -1+x+x^3 ). - _R. J. Mathar_, Oct 30 2011
%t Join[{1}, LinearRecurrence[{1, 0, 1}, {1, 1, 3}, 80]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2012 *)
%o (Haskell)
%o a179070 n = a179070_list !! (n-1)
%o a179070_list = 1 : zs where zs = 1 : 1 : 3 : zipWith (+) zs (drop 2 zs)
%o -- _Reinhard Zumkeller_, Jul 23 2012
%o (PARI) a(n)=([0,1,0; 0,0,1; 1,0,1]^(n-1)*[1;1;1])[1,1] \\ _Charles R Greathouse IV_, Apr 08 2016
%Y Cf. A000930, A029635, A097333, A214626.
%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 easy,nonn
%O 1,4
%A _Mark Dols_, Jun 27 2010
%E Simpler definition from _N. J. A. Sloane_, Aug 29 2013
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