[seqfan] Proof of conjecture needed.

[L. Edson Jeffery]

Consider the rectangular array A (https://oeis.org/draft/A257499) beginning

 1     5     9    13    17
 7    15    23    31    39
 3    19    35    51    67
27    59    91   123   155
11    75   139   203   267

A007583 and A136412 (omitting the initial 2) are bisections of of the first
column. The entry in row n and column k of A is given by

  A(n,k) = (1 + 2^n*(6*k - 3 + 2*(-1)^n))/3  (n,k >= 1).

I have a result that depends on the following

Conjecture: The rows (or columns) of A are pairwise disjoint, and their
union exhausts the odd natural numbers without duplication; or,
equivalently, the sequence A257499 (draft) is a permutation of the odd
natural numbers.

Would someone like to prove this conjecture and relay the result to me
(either here or privately)?

Ed Jeffery