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A249577 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive negative powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

%I A249577 #21 Aug 02 2024 17:44:19
%S A249577 2,-1,1,-4,3,-2,10,-7,5,-24,17,-12,58,-41,29,-140,99,-70,338,-239,169,
%T A249577 -816,577,-408,1970,-1393,985,-4756,3363,-2378,11482,-8119,5741,
%U A249577 -27720,19601,-13860,66922,-47321,33461,-161564,114243,-80782,390050,-275807,195025,-941664,665857,-470832
%N A249577 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive negative powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.
%C A249577 The sequence comprises, in reverse order, numbers to the right of a(0) in A249576.
%H A249577 Colin Barker, <a href="/A249577/b249577.txt">Table of n, a(n) for n = 0..1000</a>
%H A249577 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-2,0,0,1).
%F A249577 a(n) = -2*a(n-3)+a(n-6); G.f.: -(x^4+x^2-x+2) / (x^6-2*x^3-1). - _Colin Barker_, Nov 02 2014
%e A249577 M^-1 = [[1,-4,4][-1,3,-2][1,-2,1]]. sqrt(M[1,3]) = 2; M[3,3] = M[1,1] = -1; M[3,1] = 1. Hence a(0) = 2; a(1) = -1; a(2) = 1.
%t A249577 LinearRecurrence[{0,0,-2,0,0,1},{2,-1,1,-4,3,-2},50] (* _Harvey P. Dale_, Aug 02 2024 *)
%o A249577 (PARI) Vec(-(x^4+x^2-x+2)/(x^6-2*x^3-1) + O(x^100)) \\ _Colin Barker_, Nov 02 2014
%Y A249577 Cf. A249576, A000129, A001333, A163271, A077985.
%K A249577 sign,tabf,easy
%O A249577 0,1
%A A249577 _Russell Walsmith_, Nov 01 2014