OFFSET
1,3
COMMENTS
T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that x*(2*k+1) == 1 (mod 2^n).
The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
FORMULA
For n >= 3, T(n,k) = (2*k+1)^(2^(n-2)-1) mod 2^n, 0 <= k <= 2^(n-1) - 1.
EXAMPLE
Table starts
1,
1, 3,
1, 3, 5, 7,
1, 11, 13, 7, 9, 3, 5, 15,
1, 11, 13, 23, 25, 3, 5, 15, 17, 27, 29, 7, 9, 19, 21, 31,
1, 43, 13, 55, 57, 35, 5, 47, 49, 27, 61, 39, 41, 19, 53, 31, 33, 11, 45, 23, 25, 3, 37, 15, 17, 59, 29, 7, 9, 51, 21, 63,
...
PROG
(PARI) T(n, k) = lift(Mod(2*k+1, 2^n)^(-1))
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Jianing Song, Aug 30 2019
STATUS
approved