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A226698 Central symmetric closed knight's tour on an 8x8 board, attributed to Euler. Position after n-th move. 1
33, 50, 60, 54, 64, 47, 37, 43, 58, 52, 62, 56, 39, 45, 35, 41, 51, 57, 42, 36, 46, 40, 55, 61, 44, 34, 49, 59, 53, 63, 48, 38, 32, 15, 5, 11, 1, 18, 28, 22, 7, 13, 3, 9, 26, 20, 30, 24, 14, 8, 23, 29, 19, 25, 10, 4, 21, 31, 16, 6, 12, 2, 17, 27 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) is the position of the knight on the 8x8 board after the n-th move (a(0) gives the starting position) if one numerates the squares from left to right, top to bottom, from 1 to 64.

If the board is considered as an 8x8 matrix the square numbered N appears as element (n,m) = (floor((N-1)/8)+1, N - 8*floor((N-1)/8)), N = 1, ..., 64. Therefore, a(0) = 33, the knight's starting position is with N = 33: (5,1). The last position is with N = 27: (4,3).

a(n-1) is the inverse of A226697 read as a sequence: A226697(a(n-1)) = n, n=1, 2, ..., 64.

For the board see the example for A226697. Observe there the central symmetry with absolute difference constant 32.

REFERENCES

Martin Gardner, Mathematical Magic Show, The Math, Assoc. of Am., Washington DC, 1989, Ch. 14, Knights of the Square Table,Fig. 86, p. 191. German Translation: Mathematische Hexereien, Ullstein, 1977, Abb. 86, S. 186.

LINKS

Table of n, a(n) for n=0..63.

EXAMPLE

a(1) = 50 because after the first move the knight is on the square N = 50, or considered as matrix position at square (7, 2). The path starts at square  a(0) = 33, or (5, 1). It ends after 63 moves on square a(63) = 27, or (4, 3). The next move can close the Hamiltonian path.

CROSSREFS

Cf. A226697 (inverse).

Sequence in context: A080933 A328247 A020293 * A096278 A204381 A034815

Adjacent sequences:  A226695 A226696 A226697 * A226699 A226700 A226701

KEYWORD

nonn,fini,full

AUTHOR

Wolfdieter Lang, Jun 25 2013

STATUS

approved

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Last modified September 27 15:44 EDT 2021. Contains 347691 sequences. (Running on oeis4.)