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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.
3
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, 469, 829, 1080, 1189, 1549, 2738, 3567, 3927, 5116, 9043, 11781, 12970, 16897, 29867, 38910, 42837, 55807, 98644, 128511, 141481, 184318, 325799, 424443, 467280
OFFSET
0,7
FORMULA
Some identities:
a4(n-1) + a(4n) = a(4n+1),
a(4n) + a(4n+1) = a(4n+2),
3a(4n) = a(4n+3).
a(n) = 3*a(n-4)+a(n-8). - Colin Barker, Nov 13 2014
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
EXAMPLE
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.
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.
MATHEMATICA
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 14 204 *)
PROG
(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
(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
CROSSREFS
a(4n) = A006190
a(4n+2) = A052924.
Sequence in context: A170891 A035535 A154309 * A329301 A327134 A140514
KEYWORD
nonn,tabf
AUTHOR
Russell Walsmith, Nov 02 2014
EXTENSIONS
More terms from Colin Barker, Nov 13 2014
STATUS
approved