A375001 Square array read by antidiagonals: T(n,k) is the position of the last requested element when the elements of the k-th composition (in standard order) are requested from a self-organizing list initialized to (1, 2, 3, ...), using the move-ahead(n) updating strategy; n >= 0, k >= 1.

1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 1, 4, 1, 2, 2, 3, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 4, 1, 2, 2, 3, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 2, 3, 1, 2, 1, 1, 3, 1, 1, 2, 4, 1, 2, 2, 3, 1, 2, 1

Offset: 0

Author

Pontus von Brömssen, Jul 27 2024

Keywords

• nonn
• tabl

Comments

See A374996 for details.

Examples

			Array begins:
  n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
  ---+--------------------------------------------
   0 | 1  2  1  3  1  2  1  4  1  2  1  3  1  2  1
   1 | 1  2  1  3  2  2  1  4  1  1  1  3  2  2  1
   2 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
   3 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
   4 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
   5 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
   6 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
   7 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
   8 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
   9 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
  10 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
  11 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
  12 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
  13 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
  14 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
  15 | 1  2  1  3  2  2  1  4  2  1  1  3  2  2  1
		

Crossrefs

Cf. A007814, A025480, A374998 (row n=1), A333766, A374996, A374997.

Formula

T(0,k) = A007814(k) + 1.

T(1,k) = A374998(k).

T(n,k) = A374997(k) if n >= A333766(k)-1.

T(n,k) = A374996(n,k) - A374996(n,A025480(k-1)).

Sum_{j=1..m} T(n,k*2^j+2^(j-1)) = m*(m+1)/2 if m >= A333766(k). This is a consequence of the fact that the first m positions of the list are occupied by the elements 1, ..., m, as long as no element larger than m has been requested so far.