A375863 a(1) = 0 and a(n) = A050603(n-1)^2 for n > 0. Lexicographically earliest nonnegative sequence of integers such that the Gilbreath transform of a(1..n) gives floor(log_2(n)).

0, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 25, 25, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 36, 36, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 25

Offset: 1

Author

Thomas Scheuerle, Sep 02 2024

Keywords

• nonn

Examples

			The first row is the sequence itself. The rows below are the absolute differences
of each previous row:
0, 1, 1, 4, 4, 1, 1, 9, 9, 1, 1, 4, 4, 1, 1, 16, 16, ...
 1, 0, 3, 0, 3, 0, 8, 0, 8, 0, 3, 0, 3, 0, 15, 0, ...
  1, 3, 3, 3, 3, 8, 8, 8, 8, 3, 3, 3, 3, 15, 15, ...
   2, 0, 0, 0, 5, 0, 0, 0, 5, 0, 0, 0, 12, 0, ...
    2, 0, 0, 5, 5, 0, 0, 5, 5, 0, 0, 12, 12, ...
     2, 0, 5, 0, 5, 0, 5, 0, 5, 0, 12, 0, ...
      2, 5, 5, 5, 5, 5, 5, 5, 5, 12, 12, ...
       3, 0, 0, 0, 0, 0, 0, 0, 7, 0, ...
        3, 0, 0, 0, 0, 0, 0, 7, 7, ...
         3, 0, 0, 0, 0, 0, 7, 0, ...
          3, 0, 0, 0, 0, 7, 7, ...
           3, 0, 0, 0, 7, 0, ...
            3, 0, 0, 7, 7, ...
             3, 0, 7, 0, ...
              3, 7, 7, ...
               4, 0, ...
                4, ...
The main diagonal is floor(log_2(n)), where n = 1 in the first row and n = 2 in the second etc. .
		

Crossrefs

Cf. A050603, A039963 (Gilbreath transform of floor(log_2(n))).

Programs

PARI

a(n) = if(n == 1, 0,valuation(n-(n-2)%2, 2)^2)