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Michel Marcus: when I read the discussions, it seems that Andrew is not very happy with the definition

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Andrew Howroyd: The name states "with a move length of the current square (in decimal) + 1", but from the examples it is clear that the you mean number of digits in the decimal representation of the current square. (or perhaps more correctly the index of the current square, because the current square isn't an integer). In any case, my opinion, is that this is too obtuse/artificial. (The digital aspect in an otherwise geometric sequence seems to be just a step too far).

Andrew Howroyd: In principal I can understand the desire to generalize A316667 to a rook. In order that the rook has a possibility of becoming trapped there needs to be some constraint on the distance the rook is allowed to move. (Although really the key distinguishing merit of a rook over a knight is that isn't constrained to move a fixed distance!). However, I think the constraint needs to be a more natural choice than something derived from the decimal representation of the position in the spiral. It might be more reasonable to use either the distance from the origin or some other geometric measure.

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Joerg Arndt: Has the interest been lost?

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Joerg Arndt: Abandoned.  If anyone can help, go for it.  Otherwise we have to recycle this one.

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Peter Kagey: The name was unclear to me, in particular "move length of the current square (in decimal) + 1." I might say something like "where each move has length equal to one more than the number of digits of the current square."

Peter Kagey: Definitely crossref A055642.

Peter Kagey: Needs keyword "base".

The rook may move over squares heit has already visited, only the final square after a full move must not have been visited before.

The rook starts on square a(1) = 1. Thus its available moves are of length len(1) + 1 = 2, possibly reaching squares 11, 15, 19 and 23. Since 11 is the smallest lowestvalue, a(2) = 11. From there on , the next move has tomust have length len(11) + 1 = 3, etc.

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Cf. A316667, A323749, A336413, A336447.

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Michel Marcus: added xrefs mentioned in comments

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P. Patrick Wienhöft, <a href="https://github.com/PattuX/Traveling-Chess-Piece">Python implementation (incl. graphical output)</a>.

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Omar E. Pol: Note that there are 108 sequences in the OEIS in which the chess rook is mentioned. Could you please give us some cross-references of similar sequences?