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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 A349551 A204381 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 November 28 09:15 EST 2023. Contains 367411 sequences. (Running on oeis4.)