A249580 List of quadruples (r,s,t,u): the matrix M = [[4,12,9][2,5,3][1,2,1]] is raised to successive negative powers, then (r,s,t,u) are the square roots of M[1,3], M[1,1], M[3,3], M[3,1] respectively.

3, -1, -2, 1, -9, 4, 7, -3, 30, -13, -23, 10, -99, 43, 76, -33, 327, -142, -251, 109, -1080, 469, 829, -360, 3567, -1549, -2738, 1189, -11781, 5116, 9043, -3927, 38910, -16897, -29867, 12970, -128511, 55807, 98644, -42837

Offset

1,1

Comments

The sequence, in reverse order, comprises numbers to the left of a(0) in A249579, where the terms would be labeled a(-1), a(-2), a(-3), ... .

This sequence 'factors' into four sequences with alternating signs. Ignoring signage, they are A052906, A003688, A052924 and A006190 (listed as crossrefs below).

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Russell Walsmith, A sequence of matrices

Russell Walsmith, DCL-Chemy III: Hyper-Quadratics

Index entries for linear recurrences with constant coefficients, signature (0,0,0,-3,0,0,0,1).

Formula

a(n) = -3*a(n-4)+a(n-8). - Colin Barker, Nov 06 2014

G.f.: -x*(x^6+x^5+x^3-2*x^2-x+3) / (x^8-3*x^4-1). - Colin Barker, Nov 06 2014

Example

M^-1 = [[1,-6,9][-1,5,-6][1,-4,4]]. sqrt(M[1,3]) = 3, sqrt(M[1,1]) = -1, sqrt(M[3,3]) = -2, sqrt(M[3,1]) = 1. Then r = 3; s = -1; t = -2; ; u = 1.
M^-2 = [[16,-72,81][-12,55,-63][9,-42,49]]. sqrt(M[1,3]) = -9, sqrt(M[1,1]) = 4, sqrt(M[3,3]) = 7, sqrt(M[3,1]) = -3. Then r = -9; s = 4; t = 7; ; u = -3.

Mathematica

m[e_] := MatrixPower[{{4, 12, 9}, {2, 5, 3}, {1, 2, 1}}, -e]; g[e_, x_, y_] := (-1)^If[x == y, e, e + 1] Sqrt@ m[e][[x, y]]; f[e_] := {g[e, 1, 3], g[e, 1, 1], g[e, 3, 3], g[e, 3, 1]}; Array[f, 10] // Flatten (* Robert G. Wilson v, Dec 19 2014 *)

Prog

(PARI) Vec(-x*(x^6+x^5+x^3-2*x^2-x+3)/(x^8-3*x^4-1) + O(x^100)) \\ Colin Barker, Nov 06 2014

Crossrefs

Cf. A249579. Disregarding signage, a(4n) = A052906; a(4n+1) = A003688; a(4n+2) = A052924; a(4n+3) = A006190.

Sequence in context: A349620 A349380 A351425 * A340147 A011086 A321889

Adjacent sequences: A249577 A249578 A249579 * A249581 A249582 A249583

Keyword

sign,easy

Author

Russell Walsmith, Nov 02 2014

Status

approved