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A249579 List of quadruples (r,s,t,u): the matrix M = [[4,12,9][2,5,3][1,2,1]] 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 A249579 #26 Aug 03 2025 03:21:11
%S A249579 0,1,1,0,1,1,2,3,3,4,7,9,10,13,23,30,33,43,76,99,109,142,251,327,360,
%T A249579 469,829,1080,1189,1549,2738,3567,3927,5116,9043,11781,12970,16897,
%U A249579 29867,38910,42837,55807,98644,128511,141481,184318,325799,424443,467280
%N A249579 List of quadruples (r,s,t,u): the matrix M = [[4,12,9][2,5,3][1,2,1]] 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.
%H A249579 Colin Barker, <a href="/A249579/b249579.txt">Table of n, a(n) for n = 0..1000</a>
%H A249579 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,1).
%F A249579 Some identities:
%F A249579 a4(n-1) + a(4n) = a(4n+1),
%F A249579 a(4n) + a(4n+1) = a(4n+2),
%F A249579 3a(4n) = a(4n+3).
%F A249579 a(n) = 3*a(n-4)+a(n-8). - _Colin Barker_, Nov 13 2014
%F A249579 G.f.: -x*(3*x^6-x^5-2*x^4+x^3+x+1) / (x^8+3*x^4-1). - _Colin Barker_, Nov 13 2014
%e A249579 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])u = a(3) = 0.
%e A249579 M^2 = [[49, 126, 81][21, 55, 36][9, 24, 16]]: r = sqrt(M[3, 1]) = a(8) = 3, s = sqrt(M[3, 3]) = a(9) = 4, t = sqrt(M[1, 1]) = a(10) = 7, u = sqrt(M[1, 3]) = a(11) = 9.
%t A249579 CoefficientList[Series[- x (3 x^6 - x^5 - 2 x^4 + x^3 + x + 1) / (x^8 + 3 x^4 - 1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 14 2014 *)
%o A249579 (PARI) concat(0, Vec(-x*(3*x^6-x^5-2*x^4+x^3+x+1)/(x^8+3*x^4-1) + O(x^100))) \\ _Colin Barker_, Nov 13 2014
%o A249579 (Magma) I:=[0,1,1,0,1,1,2,3]; [n le 8 select I[n] else 3*Self(n-4)+Self(n-8): n in [1..50]]; // _Vincenzo Librandi_, Nov 14 2014
%Y A249579 Cf. A249580
%Y A249579 a(4n) = A006190
%Y A249579 a(4n+2) = A052924.
%K A249579 nonn,tabf
%O A249579 0,7
%A A249579 _Russell Walsmith_, Nov 02 2014
%E A249579 More terms from _Colin Barker_, Nov 13 2014