OFFSET
1,2
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.
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).
EXAMPLE
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
KEYWORD
nonn,tabl
AUTHOR
Bryle Morga, Sep 14 2024
STATUS
approved
