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
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jun 13 2004
EXTENSIONS
Edited and extended by Robert G. Wilson v, Jun 16 2004
STATUS
approved