Clark's Triangle

One of the properties of today's number 26618, that marks my diurnal age, is that it is a member of OEIS A100206: 

A100206

Row sums of Clark's triangle A046902: Clark's triangle: left border = 0 1 1 1..., right border = multiples of 6; other entries = sum of 2 entries above.

The setup is shown in Figure 1, although it is a mirror image of that described in OEIS A100206 because the borders are reversed. The significance of the \( (m-1)^2\) and the \(n^2\) will be explained shortly.

Figure 1: source

To quote from Figure 1's source:

Clark's triangle is a number triangle created by setting the vertex equal to 0, filling one diagonal with 1s, the other diagonal with multiples of an integer \(f\), and filling in the remaining entries by summing the elements on either side from one row above. Figure 1 above shows Clark's triangle for \(f\)=6.

Call the first column \(n\)=0 and the last column \(m=n\) so that:

$$\begin{align} c_{m \, \scriptsize{0}} &= f\,m\\
c_{m\,m} &= 1 \end{align}$$then use the recurrence relation$$c_{m\,n}=c_{m-\scriptsize{1}, \,\normalsize{n}-\scriptsize{1}}+c_{m-\scriptsize{1}, \,\normalsize{n}}$$to compute the rest of the entries. The result is given analytically by$$c_{m\,n}=f \times \binom{m} {n+1}+\binom{m-1}{n-1}$$where \( \binom{n}{k} \) is a binomial coefficient.

The interesting part is that if \(f\)=6 is chosen as the integer, then 

\( c_{m \, \scriptsize{2}} \) and \(c_{m \, \scriptsize{3}}\) simplify to$$ \begin{align} c_{m \, \scriptsize{2}} &= (m-1)^3\\c_{m \, \scriptsize{3}} &= \dfrac{(m-1)^2(m-2)^2}{4} \end{align}$$which are consecutive cubes \( (m-1)^3 \) and nonconsecutive squares$$n^2=\left ( \dfrac{(m-1)(m-2)}{2} \right )^2$$The sum of the \(m\)-th row for \(m>0\) is given by$$ \sum_{n=0}^{m} c_{m\,n}=2^{m-1}+f \times (2^{m}-1) $$ (M. Alekseyev, pers. comm., Aug. 10, 2005).

 The row sums (of which 26618 is a member and where \(f\)=6) are as follows:

0, 7, 20, 46, 98, 202, 410, 826, 1658, 3322, 6650, 13306, 26618, 53242, 106490, 212986, 425978, 851962, 1703930, 3407866, 6815738, 13631482, 27262970, 54525946, 109051898, 218103802, 436207610, 872415226, 1744830458, 3489660922

Of course there are as many sequences as there are different values of \(f\) but none is listed in the OEIS apart from A100206 where \(f\)=6. In all sequences, the ratio of successive terms rapidly approaches 2.