Cinquante signes

  Hello Math Fun, Could someone compute a few more binary terms? I am stuck with S(21). Here is the idea : We build an array T and associate to T a sequence S of binary terms that is just T read by its successive antidiagonals [in short, the term S(k) is the binary representation of the base-10 T(k)].  It turns out that this sequence S is also the upper row of T. You will understand very quickly how T was lexicographically build by distinct nonnegative terms. We start T with 0:   ..0 T is always extended by its successive antidiagonals with the smallest term not yet in T that doesn’t lead to a contradiction. ..0.....1.....a ..2.....b ..c   Now, as we want the upper row to represent T in binary, we must assign to “a” the binary value of ‘2’, which is 10 [this 10 is the 3rd term of the upper row of the array and must thus represent T(3) in binary; as 10 is 2 in binary, we assign this 10 to the letter a]: ..0.....1.....10 ..2.....b ..c   The only constraint on the ...

The idea is to fill the cells of an infinite 2D lattice grid with integers that don ’ t share any digit with their surrounding 8 cells. To do that we start with a zero somewhere and develop a square spiral S, always extending S with the smallest possible term not yet used (that doesn ’ t lead to a contradiction). Questions: will S stop at some point? Could S be infinite if some backtracking is accepted? [Hereunder is my (almost) original e-mail in French to  Carole , explaining why the above 9 is followed by 30 and 44]: >> Alors voilà, >> on enroule une spirale S autour de zéro, >> comme d’habitude, en l’étendant avec >> le plus petit terme T absent de S — de >> manière à ce qu’aucun des chiffres de T >> ne soit présent dans son voisinage V. >> Ce V est fait des 8 cases qui entourent T. >> On enroule facilement les nombres qui >> vont de 0 à 9. >> >> Mais le suivant, A, demande un calcul : >> A ne peut commenc...