OFFSET
0,6
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.
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).
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(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
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]) = 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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Russell Walsmith, Nov 03 2014
STATUS
approved