## The silver ratio and a continued fraction for log(2)

Define the three sequences,

$C_n = \sum_{k=0}^n {\binom n k} {\binom {n+k}k} = 1, 3, 13, 63, 321,\dots$

$B_n = \sum_{k=0}^n {\binom n k}^2 {\binom {n+k}k} = 1, 3, 19, 147, 1251,\dots$

$A_n = \sum_{k=0}^n {\binom n k}^2 {\binom {n+k}k}^2 = 1, 5, 73, 1445, 33001, \dots$

The last two are Apery numbers and have been discussed previously. The first are the central Delannoy numbers which obeys the limit,

$\lim_{n \to \infty} \frac{C_{n+1}}{C_n} = \left(1+\sqrt{2}\right)^2 = 5.8284\dots$

which is the square of the silver ratio.  (The ratios for the others have already been mentioned.)  These have the recurrence relations,

$n C_n = 3(2n-1)C_{n-1}-(n-1)C_{n-2}$

$n^2 B_n = (11n^2-11n+3)B_{n-1}+(n-1)^2B_{n-2}$

$n^3 A_n = (34n^3-51n^2+27n-5)A_{n-1}-(n-1)^3A_{n-2}$

To recall, the polynomials $P(n) =11n^2-11n+3$ and $P(n) = 34n^3-51n^2+27n-5$  generated numbers for the continued fractions of $\zeta(2), \zeta(3)$, so I was curious if $P(n) = 3(2n-1)$ could be used in an analogous manner.  It turns out, depending on what sign to use, it gives either $\log(2)$ or $\arctan(\frac{1}{3})$,

$\frac{1}{2}\log(2) = \cfrac{1}{v_1 - \cfrac{1^2}{v_2 - \cfrac{2^2}{v_3 - \cfrac{3^2}{v_4-\ddots}}}}$

or,

$\arctan(\frac{1}{3}) = \cfrac{1}{v_1 + \cfrac{1^2}{v_2 + \cfrac{2^2}{v_3 + \cfrac{3^2}{v_4 + \ddots}}}}$

where, starting with n = 1,

$v_n = 3(2n-1)$

This can be partly demystified since one continued fraction for the natural logarithm, and arctan, is,

$\frac{1}{2}\log\left(1+\frac{2x}{y}\right)= \cfrac{x}{1(x+y) - \cfrac{(1x)^2}{3(x+y) - \cfrac{(2x)^2}{5(x+y) - \cfrac{(3x)^2}{7(x+y) - \ddots}}}}$

and,

$\arctan(\frac{x}{z}) = \cfrac{x}{1z + \cfrac{(1x)^2}{3z + \cfrac{(2x)^2}{5z + \cfrac{(3x)^2}{7z + \ddots}}}}$

Hence, if $x+y = z$, as with the case {x, y, z} = {1, 2, 3}, then there will be identical-looking continued fractions that differ only in the signs.  But it remains interesting how the recurrence relations of these three binomial sums are involved in the continued fractions of $\log(2), \zeta(2), \zeta(3)$.  Later, we shall see there is a recurrence relation for the cfrac of $\zeta(4)$ as well.