A306309 The "zeroless Pascal triangle" read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 1, 1, 5, 1, 1, 6, 6, 2, 6, 6, 1, 1, 7, 12, 8, 8, 12, 7, 1, 1, 8, 19, 2, 16, 2, 19, 8, 1, 1, 9, 27, 21, 18, 18, 21, 27, 9, 1, 1, 1, 36, 48, 39, 36, 39, 48, 36, 1, 1, 1, 2, 37, 84, 87, 75, 75, 87, 84, 37, 2, 1

Offset: 0

Author

Rémy Sigrist, Feb 06 2019

Keywords

• nonn
• tabl
• base

Comments

Left and right edges are all 1's, interior entries are obtained by removing zeros from the sum of the two numbers above them.

For any k >= 0 and n >= 0, let d_k(n) = T(n+k, k).

For any k >= 0, d_k is eventually periodic: by induction:

- for k = 0: for any n >= 0, d_0(n) = 1, hence d_0 is 1-periodic,

- suppose that the property is true for some k >= 0,

- d_k is eventually p_k-periodic, and so d_k is bounded, say by m_k,

- d_{k+1}(n+1) - d_{k+1}(n) = d_k(n+1) <= m_k,

- so the first difference of d_{k+1} is bounded by m_k,

- A004719 has arbitrary large gaps,

and we can choose a range of m_k+1 terms that do not belong to A004719,

say x_k..x_k+m_k (with x_k > 1),

- d_{k+1}(0) = 1 < x_k,

and if d_{k+1}(n) < x_k, then d_{k+1)(n+1) < x_k,

so d_{k+1} is bounded by x_k,

- let D_{k+1}(n) = d_{k+1}(n*p_k},

- D_{k+1} is bounded,

so D_{k+1}(n + q_k) = D_{k+1}(n) for some n >= 0 and q_k > 0,

- we can assume that n*p_k is beyond the transient part of d_k,

- d_{k+1}(n*p_k + q_k*p_k + 1) = d_{k+1}(n*p_k+q_k*p_k) + d_k(n*p_k+q_k*p_k + 1)

= d_{k+1}(n*p_k) + d_k(n*p_k + 1)

= d_{k+1}(n*p_k + 1),

- we can generalize: for any m >= n*p_k, d_{k+1}(m + q_k*p_k) = d_{k+1)(m),

- and d_{k+1} is (at least q_k*p_k-)periodic, QED.

Examples

			Triangle begins:
                    1
                  1   1
                1   2   1
              1   3   3   1
            1   4   6   4   1
          1   5   1   1   5   1
        1   6   6   2   6   6   1
      1   7  12   8   8  12   7   1
    1   8  19   2  16   2  19   8   1
  1   9  27  21  18  18  21  27   9   1
...
		

Links

• Rémy Sigrist, Colored representation of the first 1000 rows (where the hue is function of T(n, k))
• Rémy Sigrist, PARI program for A306309

Crossrefs

Cf. A004719, A007318, A177274.

Programs

PARI

See Links section.

Formula

T(n, 0) = T(n, n) = 1 for n >= 0.

T(n, k) = A004719(T(n-1, k-1) + T(n-1, k)) for n >= 0 and k = 1..n-1.

T(n, 1) = A177274(n-1) for any n > 0.