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A307433
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A special version of Pascal's triangle where only powers of 2 are permitted.
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1
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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, 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, 1, 1, 16, 16, 16, 16, 1, 1, 1, 1, 1, 2, 2, 2, 1, 32, 32, 32, 1, 2, 2, 2, 1
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OFFSET
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0,5
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COMMENTS
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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.
The ones in the table form a Sierpinski gasket (A047999).
Apparently, for any k > 0, the value 2^k first occurs on row A206332(k).
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.
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2
4 4 4 4 4 4
8 8 8 8 8
16 16 16 16
32 32 32
64 64
128
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (End)
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LINKS
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EXAMPLE
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The triangle begins:
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 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 1 1 16 16 16 16 1 1 1 1
1 2 2 2 1 32 32 32 1 2 2 2 1
1 1 4 4 1 1 64 64 1 1 4 4 1 1
1 2 1 8 1 2 1 128 1 2 1 8 1 2 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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PROG
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(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))))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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