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A232007 Maximal number of moves needed by a knight to reach every square from a fixed position on an n X n chessboard, or -1 if it is not possible to reach every square. 1
0, -1, -1, 5, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 46 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

For n greater than 4 the number equals ceiling(2n/3); see A004523. - R. J. Mathar, Nov 24 2013

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Paul B. Slater, Formulas for Generalized Two-Qubit Separability Probabilities, arXiv:1609.08561 [quant-ph], 2016.

Paul B. Slater, Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure, arXiv preprint arXiv:1504.04555 [quant-ph], 2015.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

FORMULA

From Colin Barker, Apr 26 2016: (Start)

a(n) = a(n-1)+a(n-3)-a(n-4) for n>4.

G.f.: -x^2*(1-6*x^2+5*x^5-2*x^6) / ((1-x)^2*(1+x+x^2)).

(End)

EXAMPLE

For a classic 8 X 8 chessboard, a knight needs at most 6 moves to reach every square starting from a fixed position. For a 3 X 3 chessboard, it's impossible to reach the middle square starting from any other, so a(3) = -1.

PROG

(PARI) concat(0, Vec(-x^2*(1-6*x^2+5*x^5-2*x^6)/((1-x)^2*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Apr 26 2016

CROSSREFS

Sequence in context: A226578 A134206 A134209 * A019842 A303270 A244046

Adjacent sequences:  A232004 A232005 A232006 * A232008 A232009 A232010

KEYWORD

sign,easy

AUTHOR

Mateusz Szymański, Nov 16 2013

EXTENSIONS

More terms from Vaclav Kotesovec, Oct 21 2014

STATUS

approved

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Last modified October 22 10:24 EDT 2019. Contains 328317 sequences. (Running on oeis4.)