JCSBimp's Hexagonal Orthogonal

I generated the Series of Ceiling Squares systematically for each Ascent associated with yesterday’s example of m=1313. I also assembled the Beehive Plot, but won’t include that here, except to say that when the hive’s slope increased linearly, the bee line was a nice parabola.

1313 = 37^2- (8^2-(3^2 -1^2))
     = 39^2-(15^2-(5^2-(3^2-1^2)))
     = 41^2-(20^2-(6^2 -2^2))
     = 43^2-(24^2-(7^2 -3^2))
     = 45^2-(27^2-(5^2-(3^2-1^2)))
     = 47^2-(30^2 -2^2)
     = 49^2-(33^2 -1^2)
     = 51^2-(36^2-(3^2 -1^2)
     = 53^2-(39^2 -5^2)
     = 55^2-(42^2-(8^2-(4^2-2^2)))
     = 57^2-44^2.

Let’s do another one, slightly larger, where the factorization might not be so obvious to the eye, or to the series of ascents: m=11236591. This one will take many more ascents than the earlier example, more than I want to list here, but let’s look at the first thirty:

11236591 = 3354^2-(113^2 -(7^2-(3^2 -2^2)))
         = 3356^2-(163^2-(21^2-(5^2-(3^2 -1^2))))
         = 3358^2-(199^2 -(6^2-(3^2 -1^2)))
         = 3360^2-(231^2-(19^2 -3^2))
         = 3362^2-(259^2-(26^2-(7^2 -1^2)))
         = 3364^2-(283^2-(14^2-(4^2 -2^2)))
         = 3366^2-(317^2-(34^2-(6^2 -2^2)))
         = 3368^2-(327^2-(10^2 -2^2))
         = 3370^2-(347^2- 10^2)
         = 3372^2-(367^2-(30^2 -2^2))
         = 3374^2-(385^2-(31^2-(5^2 -2^2)))
         = 3376^2-(401^2  -4^2)
         = 3378^2-(419^2-(36^2-(6^2-(3^2 -1^2))))
         = 3380^2-(435^2-(38^2-(6^2-(3^2 -1^2))))
         = 3382^2-(449^2-(17^2-(5^2 -2^2)))
         = 3384^2-(465^2-(37^2 -3^2))
         = 3386^2-(479^2-(33^2-(9^2-(6^2-(3^2-1^2)))))
         = 3388^2-(493^2-(34^2-(8^2 -2^2)))
         = 3390^2-(507^2-(40^2-(8^2 -2^2)))
         = 3392^2-(519^2-(17^2 -1^2))
         = 3394^2-(533^2 -38^2)
         = 3396^2-(545^2-(29^2-(7^2-(3^2-1^2))))
         = 3398^2-(557^2-(21^2-(3^2 -2^2)))
         = 3400^2-(569^2-(19^2 -3^2))
         = 3402^2-(581^2-(24^2-(6^2-(3^2-1^2))))
         = 3404^2-(593^2 -32^2)
         = 3406^2-(605^2-(43^2-(9^2-(4^2-2^2))))
         = 3408^2-(615^2-(19^2 -3^2))
         = 3410^2-(627^2-(41^2-(9^2-(5^2-(3^2-2^2)))))
         = 3412^2-(637^2-(25^2 -3^2)) ...
         and so forth.

Does a pattern emerge? Of course it does! It is built right into the numbers; there is nowhere else it would be. But the real question is: Can we see that pattern, see the signal underneath the noise, so to speak, or look at these numbers in such a way that we could see it?