A376181 Array read by antidiagonals: Start from 1 and thereafter add gnomons of terms for each t >= 2 with width a(t-1).

1, 2, 2, 3, 2, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 3, 3, 4, 4, 5, 4, 4, 3, 4, 4, 5, 5, 5, 4, 4, 4, 4, 5, 5, 5, 5, 5, 4, 4, 4, 5, 5, 5, 6, 5, 5, 5, 4, 4, 5, 5, 5, 6, 6, 6, 5, 5, 5, 4, 5, 5, 5, 6, 6, 7, 6, 6, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 6, 6, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 6, 6, 5, 5, 5, 5

Offset: 1

Author

Bryle Morga, Sep 14 2024

Keywords

• nonn
• tabl

Comments

A gnomon is an L-shaped group of terms that are equal to each other and surrounds the top-left corner.

The sequence seems to oscillate a lot and it seems that log(a(n))/log(n) doesn't converge. However, log(max{a(i); i < n})/log(n) appears to converge to around ~0.387.

The formula a(d(d+1)/2) = n, where d = 1 + (a(1) + ... + a(n-1)), suggests that the sequence might be growing like k*n^c on average, where c = (sqrt(3)-1)/2.

The construction is similar to the Golomb sequence (A001462) in that sequence terms themselves determine repetitions, but here those repetitions are gnomon widths.

Examples

			Array begins:
      k=1 2 3 4 5 6
  n=1:  1 2 3 3 4 4
  n=2:  2 2 3 3 4 4
  n=3:  3 3 3 3 4 4
  n=4:  3 3 3 3 4 4
  n=5:  4 4 4 4 4 4
  n=6:  4 4 4 4 4 4
The first three terms by antidiagonals are 1,2,2 and they are the widths of the gnomons comprising terms 2,3,4 respectively.
		

Crossrefs

Cf. A001462, A283683.

Formula

a(d(d+1)/2) = n, where d = 1 + (a(1) + ... + a(n-1)), for n > 1.

a(n) ~ k*n^c, on average, where c = (sqrt(3)-1)/2 = 0.366... (heuristic).

max{a(i); i < n} ~ K*n^e with e ~ 0.387... (empirical).