Here is an article regarding the 3D Moore neighborhoood, regarding OEIS seq\
    uences A246031, that is the run-length transform of A246032

Note: With a few more seconds, the generating function can be rigorously pro\
    ved, and it has been.



                   2  2  2    2  2      2    2      2  2    2  2    2
On the sequence, (x  y  z  + x  y  z + x  y z  + x y  z  + x  y  + x  y z

        2  2      2          2    2  2    2      2        2      2    2
     + x  z  + x y  z + x y z  + y  z  + x  y + x  z + x y  + x z  + y  z

          2    2                2          2                 n
     + y z  + x  + x y + x z + y  + y z + z  + x + y + z + 1) ,  modulo , 2,

     evaluated at , {x = 1, y = 1, z = 1}



                              By Shalosh B. Ekhad



                    The first, 41, terms staring at n=0 are



[1, 26, 26, 124, 26, 676, 124, 1400, 26, 676, 676, 3224, 124, 3224, 1400, 10000
, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 124, 3224, 3224, 15376, 1400, 
36400, 10000, 89504, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 676]


             Just for kicks, the googol-th term of our sequence is



1578967581759084531284396199476117921426130608061678733368758435553827370086907\
27998362636561268552499200000000000000000000


                                                           i
  The first , 40, terms of the sparse subsequence at the, 2  - 1,  places are



[1, 26, 124, 1400, 10000, 89504, 707008, 5924480, 47900416, 393069824, 
3189761536, 25963397888, 210468531712, 1706090904320, 13803141607936, 
111595408530176, 901164713600512, 7271581998320384, 58625571435837952, 
472335388734974720, 3803021424555945472, 30602681612309510912, 
246127842107210007040, 1978595589150751521536, 15898858268788381120000, 
127704325127232135637760, 1025394869559837785950720, 8230701044550794151840512,
66047322008709487046884864, 529855967717721245086571264, 
4249668115993276902348001792, 34076594378530894954320190208, 
273193179174866658847566956032, 2189796492128874867606400944896, 
17549506807905482713515906898432, 140624087523724691586571442432768, 
1126662821325008405515311862025728, 9025534595958919357012676106156800, 
72293864503507265936363789228319232, 579007059512763889097172865755036416, 
4636862091411194346267051889244902912]


Using the found enumerative automaton with, 110,

    states, that we omit, it follows that

the Guessed (but absolutely certain!) rational generating function for that \
    sparse subsequence is



           10           9           8           7          6          5
- (737280 t   - 260096 t  - 887296 t  + 428928 t  + 61312 t  - 59432 t

             4         3        2              /                     10
     + 5420 t  + 1718 t  - 317 t  + 6 t + 1)  /  ((8 t - 1) (110592 t
                                             /

              9           8          7          6         5        4        3
     - 29696 t  - 140032 t  + 52992 t  + 22576 t  - 9920 t  - 856 t  + 608 t

           2
     - 17 t  - 12 t + 1))



                             and in Maple notation



-(737280*t^10-260096*t^9-887296*t^8+428928*t^7+61312*t^6-59432*t^5+5420*t^4+
1718*t^3-317*t^2+6*t+1)/(8*t-1)/(110592*t^10-29696*t^9-140032*t^8+52992*t^7+
22576*t^6-9920*t^5-856*t^4+608*t^3-17*t^2-12*t+1)


              This ends this article, that took, 0.148, seconds.