A327616 Irregular table read by rows; the first row contains a single 1; for any n > 1, row n+1 corresponds to the ordinal transform of the terms in rows 1..n.

1, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 4, 2, 1, 5, 1, 2, 3, 1, 4, 2, 1, 5, 6, 3, 2, 7, 1, 4, 8, 1, 1, 2, 3, 1, 4, 2, 1, 5, 6, 3, 2, 7, 1, 4, 8, 1, 9, 5, 3, 10, 2, 6, 11, 2, 1, 4, 7, 1, 12, 3, 1, 13, 1, 2, 3, 1, 4, 2, 1, 5, 6, 3, 2, 7, 1, 4, 8, 1, 9, 5, 3, 10, 2

Offset: 1

Author

Rémy Sigrist, Jul 06 2020

Keywords

• nonn
• look
• tabf

Comments

The ordinal transform of a sequence b(n) is the sequence t(n) = number of values in b(1), ..., b(n) which are equal to b(n).

Examples

			Triangle begins:
  1:  [1]
  2:  [1]
  3:  [1, 2]
  4:  [1, 2, 3, 1]
  5:  [1, 2, 3, 1, 4, 2, 1, 5]
  6:  [1, 2, 3, 1, 4, 2, 1, 5, 6, 3, 2, 7, 1, 4, 8, 1]
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8192

Crossrefs

Cf. A000045, A011782 (row lengths), A048297, A060138, A107946.

Programs

PARI

{ for (n=1, #a=vector(85), print1 (a[n]=if (n==1, a[n]=1, a[n]=o[a[k++]]++)", "); if (hammingweight(n)==1, k=0; o=vector(vecmax(a[1..n])))) }

Formula

Apparently:

- the greatest term in row n is A000045(n),

- a(n) = 1 iff n = 1 or n belongs to A060138,

- a(A048297(n+1)) = n (and this corresponds to the first occurrence of n),

- a(4^k) = A000045(2*k+1) for any k >= 0,

- a(2*4^k-1) = A000045(2*k+2) for any k >= 0.