

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
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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 1periodic,
 suppose that the property is true for some k >= 0,
 d_k is eventually p_kperiodic, 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(n1, k1) + T(n1, k)) for n >= 0 and k = 1..n1.
T(n, 1) = A177274(n1) 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



