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%I #12 Jun 13 2015 00:51:20
%S 1,10,63,344,1781,9030,45403,227524,1138641,5695250,28480343,
%T 142409904,712065901,3560362270,17801876883,89009515484
%N a(n) = rightmost term in M^n * [1,1,1], where M = a 3 X 3 matrix composed of the first 3 rows of A050166 (fill in the matrix with zeros): = [1 0 0 / 1 2 0 / 1 4 5].
%C A sequence relating to Catalan numbers.
%C 1. a(n)/a(n-1) tends to 5, a Catalan number. E.g. a(6)/a(5) = 45403/9030 = 4.9948... 2. Generally, with M = an N X N matrix composed of rows of A050166 (along with zeros), M^n * [1,1,1...] generates terms [a, b, c, d...] such that sequences of which a,b,c,d...are members converge upon the Catalan numbers: 1, 2, 5, 14, 42, 132...
%C Companion (M^n)[3,2] = 4*A016127(n), (M^n)[3,3] = 5^n = A000351(n), so a(n) = a(n-1) + 4*A016127(n-1) + 5^(n-1) for n>0 - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (8,-17,10).
%F Or simply with M=[1, 0, 0;1, 2, 0;1, 4, 5], a(n)=(M^n)[3, 1], (adds a leading 0 to sequence) - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005
%F G.f.: (1+2*x)/(1-8*x+17*x^2-10*x^3). [Colin Barker, Jan 31 2012]
%e a(4) = 1781 since M^4 * [1,1,1] = [1, 31, 1781].
%o (PARI) M=[1,0,0;1,2,0;1,4,5];for(i=0,10,print1((M^i)[3,1],",")) (Klasen)
%Y Cf. A050166, A016127, A000351.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Apr 18 2004
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