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%I #90 Jul 19 2019 08:48:33
%S 1,1,1,1,2,1,1,1,1,1,1,2,2,2,1,1,1,4,4,1,1,1,2,1,8,1,2,1,1,1,1,1,1,1,
%T 1,1,1,2,2,2,2,2,2,2,1,1,1,4,4,4,4,4,4,1,1,1,2,1,8,8,8,8,8,1,2,1,1,1,
%U 1,1,16,16,16,16,1,1,1,1,1,2,2,2,1,32,32,32,1,2,2,2,1
%N A special version of Pascal's triangle where only powers of 2 are permitted.
%C If the sum of the two numbers above in the triangular array is not a power of 2 (A000079), then a 1 is put in its place.
%C The ones in the table form a Sierpinski gasket (A047999).
%C Apparently, for any k > 0, the value 2^k first occurs on row A206332(k).
%C From _Bernard Schott_, May 05 2019: (Start)
%C For any m, at row 2^m - 1, there is only a string of 2^m times the number 1, then at row 2^(m+1) - 2, comes out for the first time and only once, the power of 2 equals to 2^(2^m-1). At row 2^(m+1) - 1, there are again 2^(m+1) times the number 1. This cycle can go on. Under, a part of this triangle between row 2^3 -1 and 2^4 - 2 that visualizes the explanations.
%C 1 1 1 1 1 1 1 1
%C 2 2 2 2 2 2 2
%C 4 4 4 4 4 4
%C 8 8 8 8 8
%C 16 16 16 16
%C 32 32 32
%C 64 64
%C 128
%C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (End)
%H Rémy Sigrist, <a href="/A307433/b307433.txt">Table of n, a(n) for n = 0..8255</a> (rows n = 0..127)
%H Rémy Sigrist, <a href="/A307433/a307433.png">Colored representation of the first 1024 rows</a> (where the hue is function of log(T(n,k)))
%H Rémy Sigrist, <a href="/A307433/a307433_1.png">Colored representation of the first 1024 rows</a> (where black pixels correspond to ones)
%e The triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 1 1 1
%e 1 2 2 2 1
%e 1 1 4 4 1 1
%e 1 2 1 8 1 2 1
%e 1 1 1 1 1 1 1 1
%e 1 2 2 2 2 2 2 2 1
%e 1 1 4 4 4 4 4 4 1 1
%e 1 2 1 8 8 8 8 8 1 2 1
%e 1 1 1 1 16 16 16 16 1 1 1 1
%e 1 2 2 2 1 32 32 32 1 2 2 2 1
%e 1 1 4 4 1 1 64 64 1 1 4 4 1 1
%e 1 2 1 8 1 2 1 128 1 2 1 8 1 2 1
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%o (PARI) for (r=1, 13, apply(v -> print1 (v", "), row=vector(r, k, if (k==1 || k==r, 1, hammingweight(s=row[k-1]+row[k])==1, s, 1))))
%Y Cf. A000079, A007318, A047999, A206332, A307116 (analog with Fibonacci numbers).
%K nonn,tabl,look
%O 0,5
%A _Rémy Sigrist_, May 05 2019
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