A249576 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

0, 1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 10, 12, 17, 24, 29, 41, 58, 70, 99, 140, 169, 239, 338, 408, 577, 816, 985, 1393, 1970, 2378, 3363, 4756, 5741, 8119, 11482, 13860, 19601, 27720, 33461, 47321, 66922, 80782, 114243, 161564, 195025, 275807, 390050, 470832, 665857, 941664

Offset: 0

Author

Russell Walsmith, Nov 01 2014

Keywords

• nonn
• tabf
• easy

Comments

Numbers to the left of a(0) are in A249577.

Some identities:

a(3n - 2) + a(3n - 1) = a(3n + 1).

a(3n) + a(3n + 1) = a(3(n + 1)).

a(3n - 2) + a(3n + 1) = a(3n + 2).

a(3n) + a(3n - 1) + a(3(n - 2)) = a(3n + 1).

a(3n - 1)a(3n) + a(3n + 2)a(3(n + 1)) = a(6n + 2).

Examples

			M^0 = the 3 X 3 identity matrix = [[1,0,0][0,1,0][0,0,1]]. M[3,1] = 0; M[1,1] = 1; M[1,3] = 0. So the first triple is r = a(0) = 0; s = a(1) = 1; t = a(2) = 0.
M^1 = [[1,4,4][1,3,2][1,2,1]], so r = a(3) = 1; s = a(4) = 1; t = a(5) = 2.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,1).

Crossrefs

a(3n) = the n-th term of A000129, the Pell numbers.

a(3n+1) = n-th term of A001333.

a(3n+2) = n-th term of A163271.

Programs

Mathematica

LinearRecurrence[{0,0,2,0,0,1},{0,1,0,1,1,2},60] (* Harvey P. Dale, Dec 29 2021 *)

PARI

concat(0, Vec(-x*(2*x^4-x^3+x^2+1)/(x^6+2*x^3-1) + O(x^100))) \\ Colin Barker, Nov 02 2014

Formula

a(n) = -2*a(n-3)+a(n-6); G.f.: -x*(2*x^4-x^3+x^2+1) / (x^6+2*x^3-1). - Colin Barker, Nov 02 2014