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A094895 Sequence generated from a Knight's tour of a 4 X 4 chessboard considered as a matrix. 2

%I

%S 1,280,8524,295840,10014256,340831360,11585508544,393929320960,

%T 13393420731136,455377714186240,15482831007960064,526416344465121280,

%U 17898154990259286016,608537275441252433920,20690267318823093059584

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

%C 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.

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

%H G. C. Greubel, <a href="/A094895/b094895.txt">Table of n, a(n) for n = 1..650</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (24,324,544).

%F 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.

%F From _Colin Barker_, Oct 21 2012: (Start)

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

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

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

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

%t 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 *)

%t Table[2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1)), {n,20}] (* _G. C. Greubel_, Jul 11 2019 *)

%o (PARI) vector(20, n, 2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1))) \\ _G. C. Greubel_, Jul 11 2019

%o (Magma) [2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1)): n in [1..20]]; // _G. C. Greubel_, Jul 11 2019

%o (Sage) [2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1)) for n in (1..20)] # _G. C. Greubel_, Jul 11 2019

%o (GAP) List([1..20], n-> 2^(n-2)*((-4)^n + 9*(-1)^n + 15*17^(n-1))); # _G. C. Greubel_, Jul 11 2019

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Jun 13 2004

%E Edited and extended by _Robert G. Wilson v_, Jun 16 2004

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Last modified February 5 08:15 EST 2023. Contains 360082 sequences. (Running on oeis4.)