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.