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%I #33 Jan 02 2023 12:30:48
%S 1,0,0,1,3,8,22,61,169,468,1296,3589,9939,27524,76222,211081,584545,
%T 1618776,4482864,12414361,34378995,95205488,263651830,730128997,
%U 2021940649,5599344780,15506222688,42941263933,118916913891,329315700428,911971451326,2525515567441
%N Expansion of (-x^2 + 3*x - 1)/(x^3 - x^2 + 3*x - 1).
%C _Peter A. Lawrence_ (see links) has posted a challenge to find a 3x3 integer matrix with "smallish" elements whose powers generate a sequence that is not in the OEIS. This sequence is one of the solutions found.
%C a(n+3) is the number of ternary strings of length n in which the number of substrings of the form 0011 equals the number of substrings of the form 11. - _John M. Campbell_, Nov 02 2013
%H Alois P. Heinz, <a href="/A200752/b200752.txt">Table of n, a(n) for n = 0..700</a>
%H Peter Lawrence et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2011-November/008535.html">sequence challenge</a> and follow-up messages on the SeqFan list, Nov 21 2011
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,1)
%F G.f.: (-x^2+3*x-1)/(x^3-x^2+3*x-1).
%F Term (1,1) in the 3x3 matrix [0,1,0; 0,0,1; 1,-1,3]^n.
%F a(n) = 3*a(n-1) -a(n-2) +a(n-3) with a(0)=1, a(1)=0, a(2)=0. - _Taras Goy_, Jul 23 2017
%p a:= n-> (<<0|1|0>, <0|0|1>, <1|-1|3>>^n)[1, 1]:
%p seq(a(n), n=0..50);
%Y Cf. A200676, A200739, A200715.
%K nonn,easy
%O 0,5
%A _Alois P. Heinz_, Nov 21 2011
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