By Douglas Hofstadter and Doron Zeilberger

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)

Written: Jan. 7, 2014 ; Revised version (with first-named author added): April 17, 2014.

In our recent work on an interesting enumeration problem, we got "stuck". It all boils down to proving a certain, seemingly simple, determinant evaluation. This is a challenge for all you determinant-evaluation whizes, and an opportunity to make the OEIS richer, since we are pledging 500 dollars for a proof and 50 dollars for a disproof (in honor of the prover/disprover, with their name mentioned!)

Added Jan. 9, 2014: we posted this in the arxiv and got quite a few attempted proofs, that unfortunately turned out to be wrong. Not surprisingly, since we `lied' when we said that "it came up in an enumeration problem". The conjectured explicit determinant evaluation is a disguised form of "one half" of the notorious Collatz problem, i.e. that there are no finite orbits. Congratulations to those people who figured it out, including Robin Chapman, who kindly permitted us to post his proof of equivalence.

The input file produces the output file

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger