A094944 A sequence with a(n)/a(n-1) converging to 7, generated from a semi-magic square.

1, 17, 121, 769, 5681, 39121, 274345, 1922945, 13447009, 94165777, 659108825, 4613711233, 32296542097, 226073894609, 1582520918281, 11077645104385, 77543495432897, 542804558486545, 3799631689665337, 26597422073425409

Offset: 1

Author

Gary W. Adamson, May 25 2004

Keywords

• nonn
• easy

Comments

3 rows: 1 4 2, 2 1 4, 4 2 1 form a semi-magic square: row sums and columns and the diagonal = 7, the convergent of the sequence.

Examples

			a(4) = 769 since M^4 * [1 0 0] = [769 824 808].
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,21,49).

Crossrefs

Cf. A094943 uses the same format and operations but with different terms.

Programs

Mathematica

a[n_] := (MatrixPower[{{1, 4, 2}, {2, 1, 4}, {4, 2, 1}}, n].{{1}, {0}, {0}})[[1, 1]]; Table[ a[n], {n, 10}] (* Robert G. Wilson v, May 29 2004 *)

Formula

Let M = the 3 X 3 matrix [1 4 2 / 2 1 4 / 4 2 1], then with M^n * [1 0 0] = [p q r], a(n) = p.

G.f.: -x*(7*x+1)^2 / ((7*x-1)*(7*x^2+4*x+1)). [Colin Barker, Dec 06 2012]

3*a(n) = 7^n +2 *(-1)^n *A213421(n). - R. J. Mathar, Nov 15 2019

Extensions

Edited and extended by Robert G. Wilson v, May 29 2004