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Cf. A242511, A242512, A242514, A242590A183043, A242591.

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a(n) = maximal number of shortest knight's move paths, from origin at center of an infinite open chess boardchessboard, to any square within n moves.

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Jon E. Schoenfield: Joerg -- I could be mistaken on this (even though I'm a native speaker of American English, and I think I do okay at writing it, most of the time <g>), but here's how I see it:

Constructions like

     A is replaced by B
or
     A was replaced by B

are okay if B is the entity doing the replacing.  (The passive voice is used in both of the above examples, and those who urge that active voice be used wherever possible would prefer

     B replaces A
or
     B replaced A

... but I'm not bothered by the use of passive voice.  [Or should I say "the use of passive voice does not bother me"?])  ;-)

However, when there's a third entity doing the replacing, I think it's a different situation.  If an explicit reference is made to that third entity, e.g., in

    I replaced A ___ B
or
    He should replace A ___ B

... I can't remember ever having seen "by" used in the blank; I've seen only the word "with" used in constructions like those.

Now, it's true that the sentence at hand in the Comments section doesn't explicitly refer to an entity that's doing the replacing.  But (and, again, I could be mistaken) I think the same rule applies, and

     The same sequence results after replacing A with B.

seems correct to me, while

     The same sequence results after replacing A by B.

seems incorrect.

Perhaps complicating the situation is the fact that what I conveniently replaced above with the letter A is a phrase that includes the word "within" ... and it may be that the presence of that word so close to 'with' makes

     replacing 'within n moves' with

seem a little awkward.

Anyway, that's the way I see it, but if you and/or others see it differently, I won't complain if you decide to replace my "with" _with_ "by" ...    ;-)

Thanks for your time in reading this blathering pink-box comment!  -- Jon

For n=5, there are 100 shortest paths of length 5 steps from (0,0) to (7,0);); no square at 5 (or fewer) moves from the origin has more shortest paths.

no square at 5 (or fewer) moves from the origin has more shortest paths.

easy,nonn,more,walk,new

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Michel Marcus: I think more keyword not needed any longer.

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Joerg Arndt: "replace by" looks neater to me (non-native) than "replace with".

The same sequence results after replacing ` 'within n moves' by `with 'at shortest distance n moves'.

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