A348648 Triangle T(n, k), n >= 0, 0 <= k <= n, read by rows; the even terms of Pascal's triangle (A007318) are clustered as wXwXw subtriangles with w = 2^s-1 for some s > 0; if A007318(n, k) is even then T(n, k) gives the corresponding s otherwise T(n, k) = 0.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 3, 3, 3, 3, 0, 0, 0, 3, 3, 3, 3, 3, 3, 0, 0, 0, 1, 0, 3, 3, 3, 3, 3, 0, 1, 0, 0, 0, 0, 0, 3, 3, 3, 3, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 3, 3, 0, 2, 2, 2, 0

Offset: 0

Author

Rémy Sigrist, Oct 27 2021

Keywords

• nonn
• look
• tabl

Comments

It is possible to build this triangle with the following procedure:

- T_0 is the 2X2X2 triangle with all 0's:

^

T_0 = /0\

/0 0\

+-----+

- for s > 0, T_s is obtained by arranging 3 copies of T_{s-1} and one wXwXw triangle (where w=2^s-1) will all s's as follows:

^

/ \

/ \

/ \

/T_{s-1}\

T_s = +---------+

/ \s ... s/ \

/ \. ./ \

/ \. ./ \

/T_{s-1}\s/T_{s-1}\

+---------+---------+

- the triangle {T(n, k)} is the limit of T_s as s tends to infinity.

For any n >= 0, A001316(n) gives the number of 1's in row 4*n + 2 (and there are no 1's elsewhere).

Examples

			Triangle T(n, k) begins (with dots instead of 0's):
                    .
                   . .
                  . 1 .
                 . . . .
                . 2 2 2 .
               . . 2 2 . .
              . 1 . 2 . 1 .
             . . . . . . . .
            . 3 3 3 3 3 3 3 .
           . . 3 3 3 3 3 3 . .
          . 1 . 3 3 3 3 3 . 1 .
         . . . . 3 3 3 3 . . . .
        . 2 2 2 . 3 3 3 . 2 2 2 .
       . . 2 2 . . 3 3 . . 2 2 . .
      . 1 . 2 . 1 . 3 . 1 . 2 . 1 .
     . . . . . . . . . . . . . . . .
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8255 (rows for n = 0..127, flattened)
• Rémy Sigrist, Colored representation of the first 2^9 rows (black pixels correspond to 0's)
• Rémy Sigrist, Colored logarithmic scatterplot of A014428 (where the colors are given by the present sequence)
• Rémy Sigrist, Colored logarithmic scatterplot of A007318 (where the colors are given by the present sequence, and black pixels denotes 0's)
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A001316, A007318, A047999, A014428, A129760.

Programs

PARI

T(n,k) = { if (n<=1, return (0)); my (s=#binary(n)-1); n-=2^s; if (k<=n, return (T(n,k)), k<2^s, return (s), return (T(n,k-2^s))) }

Formula

T(n, k) = 0 iff A047999(n, k) = 1.

2^T(n, 0) OR 2^T(n, 1) OR ... OR 2^T(n, n) = 1 + A129760(n+1) (where OR denotes the bitwise OR operator).