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.

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

Author

Russell Walsmith, Nov 03 2014

Keywords

• nonn
• easy
• tabf

Comments

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.

Examples

			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.
		

Links

• 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).

Crossrefs

Cf. A249579, A249580, A147722, A052984.

Programs

Mathematica

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

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

Formula

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