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 A094895 Sequence generated from a Knight's tour of a 4 X 4 chessboard considered as a matrix. 2
 1, 280, 8524, 295840, 10014256, 340831360, 11585508544, 393929320960, 13393420731136, 455377714186240, 15482831007960064, 526416344465121280, 17898154990259286016, 608537275441252433920, 20690267318823093059584 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 6 18:58 EST 2022. Contains 358644 sequences. (Running on oeis4.)