2,6

The triangle begins with (2, 2).

Each row can be produced from the previous row by adding one to each number and resetting to zero any which would equal their column number. A number p > 2 is prime iff row p contains no zeros.

A new column k begins at row n when n is a perfect square. T(n, k) is then 1, while T(n, sqrt(n) = k - 1) is 0.

Zeros correspond to ones in the Redheffer matrix. Various interesting patterns exist. For example, as noted above, T(n^2, n) = 0. Also:

T(n^2 + n, n) = T(n^2 + n, n + 1) = 0

T(n^2 + n - 2, n - 1) = 0

T(n^2 - 1, n - 1) = 0

For all k in some [0, c]:

T(n^2, 2 + k) = 0 if n is even

T(n^2, 2 + k) = 1 if n is odd

T(n^2 + n, 2 + k) = 0

Every zero is located on some parabola directed toward n = 0, having either even width and produced by an even sequence; or having an odd width and produced by an odd sequence. In either case, the relevant sequence has constant first differences 2. T(n^2, n) begins an odd parabola, while T(n^2 + n, n) begins an even parabola and parabolas of either variety extend from infinitely many other locations.

The triangle begins:

0

1

0 1

1 2

0 0

1 1

0 2

1 0 1

0 1 2

1 2 3

0 0 0

1 1 1

0 2 2

1 0 3

0 1 0 1

1 2 1 2

0 0 2 3

1 1 3 4

0 2 0 0

1 0 1 1

0 1 2 2

1 2 3 3

0 0 0 4

Eric Weisstein's World of Mathematics, Redheffer Matrix

Cf. A002321, A083058, A144912

Sequence in context: A023565 A025922 A151843 this_sequence A001842 A029429 A064559

Adjacent sequences: A147693 A147694 A147695 this_sequence A147697 A147698 A147699

easy,nonn,tabf

Reikku Kulon (reikku(AT)gmail.com), Nov 10 2008