1,2

The 4 X 4 chessboard format is a Knight's tour (inclusive of the integers 1-16) as shown on p. 76 of Watkins, which he generated from the Gray code. a(n)/a(n-1) tends to 34, an eigenvalue of the characteristic polynomial of the matrix: x^4 - 24x^3 - 324x^2 - 544x. The recursion multipliers (24), (324) and (544) may be seen with changed signs as the 3 rightmost coefficients of the characteristic polynomial.

John J. Watkins, "Across the Board, The Mathematics of Chessboard Problems" Princeton University Press, 2004, p. 76.

Begin with the 4 X 4 matrix M = [1 6 15 12 / 14 9 4 7 / 5 2 11 16 / 10 13 8 3]. Then a(n) = leftmost term in M^n * [1 0 0 0]. Recursion method: a(n+3) = 24*a(n+2) + 324*a(n+1) + 544*a(n); n>4.

a(3) = 8524, leftmost term of M^3 * [1 0 0 0]: [8524, 8816, 8780, 8560].

a(5) = 10014256 = 24*295840 + 324*8524 + 544*280.

a[n_] := (MatrixPower[{{1, 6, 15, 12}, {14, 9, 4, 7}, {5, 2, 11, 16}, {10, 13, 8, 3}}, n].{{1}, {0}, {0}, {0}})[[1, 1]]; Table[ a[n], {n, 15}] (from Robert G. Wilson v Jun 16 2004)

Sequence in context: A038656 A160116 A024214 this_sequence A091034 A142444 A139655

Adjacent sequences: A094892 A094893 A094894 this_sequence A094896 A094897 A094898

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2004

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 16 2004