I have been studying a pair of recursively defined integer sequences that alternate between perfect squares (1’s) and pronic numbers (0’s):

Ascending Nitya sequence (starting from 1):

a(1) = 1, \quad a(n+1) = a(n) + \lceil \sqrt{a(n)} \rceil

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, \dots

Descending Nitya sequence (starting from ):

b(1) = m^2, \quad b(n+1) = b(n) - \lfloor \sqrt{b(n)} \rfloor

100, 90, 81, 72, 64, 56, \dots, 0, 0, 0, \dots

Key observation:

Odd indices = squares (“1’s”)

Even indices = pronics (“0’s”)

In the descending version, the sequence eventually collapses to 0, which then holds stable context.

This suggests a metaphor: 1’s carry contextual information about 0’s, and 0’s carry contextual information about 1’s.

My question: Could such a recursive, context-storing alternation between 1’s and 0’s offer insights for:

designing LLM architectures, context windows and memory compression?

I would love to hear if anyone sees analogies with existing work in attention mechanisms, recursive memory, or symbolic encoding.