A249579 -id:A249579 - cp's OEIS frontend

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

Russell Walsmith, Nov 02 2014

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).

			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.

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).

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

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 *)

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

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

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.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 3, 4, 5, 8, 13, 20, 23, 36, 59, 92, 105, 164, 269, 420, 479, 748, 1227, 1916, 2185, 3412, 5597, 8740, 9967, 15564, 25531, 39868, 45465, 70996, 116461, 181860, 207391, 323852, 531243, 829564, 946025, 1477268, 2423293, 3784100

Offset: 0

Russell Walsmith, Nov 03 2014

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.

Iff |r * u - s * t| = 1 then terms to the left of a(0) are all integers.

			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.
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.

Colin Barker, Table of n, a(n) for n = 0..1000
Russell Walsmith, DCL-Chemy III: Hyper-Quadratics
Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-2).

Cf. A249579, A249580, A147722, A052984.

LinearRecurrence[{0,0,0,5,0,0,0,-2},{0,1,1,0,1,2,3,4},50] (* Harvey P. Dale, Aug 01 2016 *)

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

a(4n) + a(4n + 1) = a(4n + 2).
a(4n + 1) + a(4n + 2) + a(4n + 3) - a(4n) = a(4n + 5)
4a(4n) = a(4n+3).
a(4n+1) = A147722(n), a(4n+2) = A052984(n).
a(n) = 5*a(n-4)-2*a(n-8). - Colin Barker, Nov 04 2014
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

cp's OEIS Frontend

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.

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