A329278 Irregular table read by rows. The n-th row is the permutation of {0, 1, 2, ..., 2^n-1} given by T(n,k) = k(k+1)/2 (mod 2^n).

0, 0, 1, 0, 1, 3, 2, 0, 1, 3, 6, 2, 7, 5, 4, 0, 1, 3, 6, 10, 15, 5, 12, 4, 13, 7, 2, 14, 11, 9, 8, 0, 1, 3, 6, 10, 15, 21, 28, 4, 13, 23, 2, 14, 27, 9, 24, 8, 25, 11, 30, 18, 7, 29, 20, 12, 5, 31, 26, 22, 19, 17, 16, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 2

Offset

0,6

Comments

Conjecture: for n > 0, the n-th row has 2^(n-1)-1 descents.

T(n,k) = A000217(k) for 0 <= k <= A017911(n+1), and T(n,2^n-1) = 2^(n-1).

Using T(n,k) we always can construct semimagic square of size 2^n X 2^n for n >= 0 (with a single exception at n = 1). To do this, take a square matrix M of size 2^n X 2^n and apply M(i,j) = 1 + ((j-i) mod 2^n)*2^n + [(j-i) even]*s(n,i) + [(j-i) odd]*(2^n - s(n,i) - 1) for 0 <= i,j < 2^n where T(n,s(n,k)) = k (see Jack Edward Tisdell link). - Mikhail Kurkov, Feb 06 2026

Links

Peter Kagey, Table of n, a(n) for n = 0..8190 (first 12 rows)

Jack Edward Tisdell, Simple method to construct semimagic square of size 2^m X 2^m, answer to question on MathOverflow, 2026.

Example

Table begins:
  0;
  0, 1;
  0, 1, 3, 2;
  0, 1, 3, 6,  2,  7, 5,  4;
  0, 1, 3, 6, 10, 15, 5, 12, 4, 13, 7, 2, 14, 11, 9, 8;
  ...

Maple

T:= (n, k)-> irem(k*(k+1)/2, 2^n):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);  # Alois P. Heinz, Jan 08 2020

Crossrefs

Cf. A000217, A000225, A017911, A053645, A105332, A330766, A331105, A363674.

T(n,2^n-1) gives A131577.

Sequence in context: A151844 A286223 A341163 * A382467 A348966 A008783

Adjacent sequences: A329275 A329276 A329277 * A329279 A329280 A329281

Keyword

nonn,tabf,look

Author

Peter Kagey, Nov 11 2019

Status

approved