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A249581 List of quadruples (r,s,t,u): the matrix M = [[9,24,16][3,10,8][1,4,4]] is raised to successive powers, then (r,s,t,u) are the square roots of M[3,1], M[3,3], M[1,1], M[1,3] respectively.

%I A249581 #20 Aug 01 2016 14:00:05
%S A249581 0,1,1,0,1,2,3,4,5,8,13,20,23,36,59,92,105,164,269,420,479,748,1227,
%T A249581 1916,2185,3412,5597,8740,9967,15564,25531,39868,45465,70996,116461,
%U A249581 181860,207391,323852,531243,829564,946025,1477268,2423293,3784100
%N A249581 List of quadruples (r,s,t,u): the matrix M = [[9,24,16][3,10,8][1,4,4]] is raised to successive powers, then (r,s,t,u) are the square roots of M[3,1], M[3,3], M[1,1], M[1,3] respectively.
%C A249581 The general form of these matrices is [[t^2,2tu,u^2][rt,st+ru,su][r^2,2rs,s^2]]. Different symmetries have different properties.
%C A249581 Iff |r * u - s * t| = 1 then terms to the left of a(0) are all integers.
%H A249581 Colin Barker, <a href="/A249581/b249581.txt">Table of n, a(n) for n = 0..1000</a>
%H A249581 Russell Walsmith, <a href="/A249581/a249581.pdf">DCL-Chemy III: Hyper-Quadratics</a>
%H A249581 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,5,0,0,0,-2).
%F A249581 a(4n) + a(4n + 1) = a(4n + 2).
%F A249581 a(4n + 1) + a(4n + 2) + a(4n + 3) - a(4n) = a(4n + 5)
%F A249581 4a(4n) = a(4n+3).
%F A249581 a(4n+1) = A147722(n), a(4n+2) = A052984(n).
%F A249581 a(n) = 5*a(n-4)-2*a(n-8). - _Colin Barker_, Nov 04 2014
%F A249581 G.f.: x*(4*x^6-2*x^5-3*x^4+x^3+x+1) / (2*x^8-5*x^4+1). - _Colin Barker_, Nov 04 2014
%e A249581 M^0 = [[1,0,0][0,1,0][0,0,1]]. r = sqrt(M[3,1]) = a(0) = 0; s = sqrt(M[3,3]) = a(1) = 1; t = sqrt(M[1,1]) = a(2) = 1; u = sqrt(M[1,3]) = a(3) = 0.
%e A249581 M^1 = [[9,24,16][3,10,8][1,4,4]]. r = sqrt(M[3,1]) = a(4) = 1; s = sqrt(M[3,3]) = a(5) = 2; t = sqrt(M[1,1]) = a(6) = 3; u = sqrt(M[1,3]) = a(7) = 4.
%t A249581 LinearRecurrence[{0,0,0,5,0,0,0,-2},{0,1,1,0,1,2,3,4},50] (* _Harvey P. Dale_, Aug 01 2016 *)
%o A249581 (PARI) concat(0, Vec(x*(4*x^6-2*x^5-3*x^4+x^3+x+1)/(2*x^8-5*x^4+1) + O(x^100))) \\ _Colin Barker_, Nov 04 2014
%Y A249581 Cf. A249579, A249580, A147722, A052984.
%K A249581 nonn,easy,tabf
%O A249581 0,6
%A A249581 _Russell Walsmith_, Nov 03 2014