Codon Anti-codon Magic Square

 CODON ANTI-CODON MAGIC SQUARE

  The original name was: "The sequence is an anti-diagonal of the decimal of a mapped 4-ary Gray code matrix as a triangular sequence."

  Gary W. Adamson's explanation of the sequence: Here's the conversion rules for the codons, 4-Ary gray code, which "turns out" to be the most appropriate format for mapping the Codons on a gray code Karnaugh map. The "why" this is the appropriate format relates to a degree of trial and error to find the proper fit in terms of the numbers of hydrogen bonds per codon- anticodon.

(Antti Karttunen's comment: obscure definition. The "degree of trial and error" should be defined transparently.)

  The "H-bond codon-anticodon magic square" map by Gary Adamson, published on page 287 of Cliff Pickover's book "Zen of Magic Squares..." looks like this:
  

CCC CCU CUU CUC UUC UUU UCU UCC

CCA CCG CUG CUA UUA UUG UCG UCA

CAA CAG CGG CGA UGA UGG UAG UAA

CAC CAU CGU CGC UGC UGU UAU UAC

AAC AAU AGU AGC GGC GGU GAU GAC

AAA AAG AGG AGA GGA GGG GAG GAA

ACA ACG AUG AUA GUA GUG GCG GCA

ACC ACU AUU AUC GUC GUU GCU GCC

Using the conversion rules:
0 = C, 1 = A, 2 = G, 3 = U,
we convert to 4-ary gray code:
 

000 003 033 030 330 333 303 300

001 002 032 031 331 332 302 301

011 012 022 021 321 322 312 311

010 013 023 020 320 323 313 310

110 113 123 120 220 223 213 210

111 112 122 121 221 222 212 211

101 102 132 131 231 232 202 201

100 103 133 130 230 233 203 200

To convert back to decimal:
 

+0 +3 14 15 58 57 62 63

+1 +2 13 12 59 56 61 60

+6 +7 +8 11 54 55 50 49

+5 +4 +9 10 53 52 51 48

26 25 30 31 32 35 46 47

27 24 29 28 33 34 45 44

22 23 18 17 38 39 40 43

21 20 19 16 37 36 41 42

... and that's it! Notice how the 1,2,3... jumps around, somewhat like a Peano curve, from one 4-unit cell to the next.

  Antti Karttunen's notes: The steps 1 & 2 are clear, but the step 3 would not produce the array given here, but instead the array A163239. Furthermore, in Pickover's book the conversion rules C=0, A=1, U=2, G=3 are used, in which case we get the array A163235. 

 

Also, the path taken by the terms does not form a continuous Peano curve (Hamiltonian path), because there are discontinuities, e.g. when going from 3 to 4, or from 15 to 16. See A163357/A163359 & A163334/A163336 for examples of continuous Peano/Hilbert curves/paths in an NxN grid. However, this sequence is uniquely defined by the formula a(n) = A163485(A057300(A054238(n))). The 8x8 array given at the step 3 is the top left corner of the infinite square array whose antidiagonal gives this sequence.

REFERENCES

Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002, pp. 285-289.

 

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