

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
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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 TwoQubit Separability Probabilities, arXiv:1609.08561 [quantph], 2016.
Paul B. Slater, Hypergeometric/DifferenceEquationBased Separability Probability Formulas and Their Asymptotics for Generalized TwoQubit States Endowed with Random Induced Measure, arXiv preprint arXiv:1504.04555 [quantph], 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(n1)+a(n3)a(n4) for n>4.
G.f.: x^2*(16*x^2+5*x^52*x^6) / ((1x)^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*(16*x^2+5*x^52*x^6)/((1x)^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



