A307433 A special version of Pascal's triangle where only powers of 2 are permitted.

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

Offset: 0

Author

Rémy Sigrist, May 05 2019

Keywords

• nonn
• tabl
• look

Comments

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).

From Bernard Schott, May 05 2019: (Start)

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)

Examples

			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
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8255 (rows n = 0..127)
• Rémy Sigrist, Colored representation of the first 1024 rows (where the hue is function of log(T(n,k)))
• Rémy Sigrist, Colored representation of the first 1024 rows (where black pixels correspond to ones)

Crossrefs

Cf. A000079, A007318, A047999, A206332, A307116 (analog with Fibonacci numbers).

Programs

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))))