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A306309 The "zeroless Pascal triangle" read by rows. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

LINKS

Table of n, a(n) for n=0..77.

Rémy Sigrist, Colored representation of the first 1000 rows (where the hue is function of T(n, k))

Rémy Sigrist, PARI program for A306309

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.

EXAMPLE

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

...

PROG

(PARI) See Links section.

CROSSREFS

Cf. A004719, A007318, A177274.

Sequence in context: A242312 A140279 A096145 * A123264 A034930 A095142

Adjacent sequences:  A306306 A306307 A306308 * A306310 A306311 A306312

KEYWORD

nonn,tabl,base

AUTHOR

Rémy Sigrist, Feb 06 2019

STATUS

approved

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Last modified December 12 09:36 EST 2019. Contains 329953 sequences. (Running on oeis4.)