A094895 Sequence generated from a Knight's tour of a 4 X 4 chessboard considered as a matrix.

1, 280, 8524, 295840, 10014256, 340831360, 11585508544, 393929320960, 13393420731136, 455377714186240, 15482831007960064, 526416344465121280, 17898154990259286016, 608537275441252433920, 20690267318823093059584

Offset

1,2

Comments

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.

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..650

Index entries for linear recurrences with constant coefficients, signature (24,324,544).

Formula

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.

From Colin Barker, Oct 21 2012: (Start)

a(n) = 2^(n-2)*(17*(-4)^n + 153*(-1)^n + 15*17^n)/17.

G.f.: x*(1 +256*x +1480*x^2)/((1+2*x)*(1+8*x)*(1-34*x)). (End)

Example

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.

Mathematica

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, 20}] (* Robert G. Wilson v, Jun 16 2004 *)
Table[2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1)), {n, 20}] (* G. C. Greubel, Jul 11 2019 *)

Prog

(PARI) vector(20, n, 2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1))) \\ G. C. Greubel, Jul 11 2019
(Magma) [2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1)): n in [1..20]]; // G. C. Greubel, Jul 11 2019
(Sage) [2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1)) for n in (1..20)] # G. C. Greubel, Jul 11 2019
(GAP) List([1..20], n-> 2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1))); # G. C. Greubel, Jul 11 2019

Crossrefs

Sequence in context: A297724 A024214 A357248 * A223107 A218411 A272715

Adjacent sequences: A094892 A094893 A094894 * A094896 A094897 A094898

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 13 2004

Extensions

Edited and extended by Robert G. Wilson v, Jun 16 2004

Status

approved