Integer Sequences Related To π 

For any given integers s_{0} and s_{1} we can construct the infinite sequence s_{n}, n = 0, 1, 2, ... generated by the recurrence 



For example, with the initial values s_{0} = 0, s_{1} = 1 we produce the sequence 



On the other hand, if we take the initial values s_{0} = 1, s_{1} = 0 we produce the sequence 



For convenience, let s_{k}(m,n) denote s_{k} based on the initial values s_{0} = m and s_{1} = n. The two sequences listed above can serve as the "basis" for all sequences of this form, according to the relation 



The ratios of corresponding terms in these sequences approach some interesting values. For example, we can show that for any integers m,n (positive or negative) 



Simply dividing two general expressions of this form, we get the more general relation for any w,x,y,z 



This applies to any real values of w,x,y,z, not just integers, so it gives the limit of ratios of terms of any sequence s_{k}(x,y) to terms of the sequence s_{k}(w,z). To illustrate, note that the sequence s_{k}(1,1) has the values 



and the sequence s_{k}(2,1) has the values 



The ratio of the 9th terms of these sequences is 



If we add 2 to this ratio and multiply by 4/3 we get 3.141593... Another interesting observation concerns the growth rate of the individual sequences. It appears that the difference in consecutive ratios of consecutive values approaches 1 + √2. In other words, for any given initial values, we have 



It would be interesting to determine an explicit solution of the recurrence (1), which is a simple homogeneous linear 2ndorder recurrence, but with nonconstant coefficients. 

Since (3) shows that all the sequences are easily related, it's enough to consider just the ratio of two sequences. Define the integer sequences D_{k} and N_{k} such that both satisfy the recurrence relation (1) with the initial values N_{0} = 0, N_{1} = 1 and D_{0} = 1, D_{1} = 1. Thus the values of N and D are 



It can be shown that the ratio N_{k}/D_{k} equals the kth convergent of the continued fraction for the inverse tangent of 1: 



Incidentally, this continued fraction can also be inferred from the fact that 



The continued fraction can be converted into the following infinite series 



where the kth term c_{k} is just the difference between the (k–1)th and the (k+1)th convergent of the continued fraction. In other words 



As a result, the partial sums of the series are just the sums of two consecutive convergents of the continued fraction. Clearly the sequence of numerators is just the odd numbers. Is there a simple way of characterizing the denominators? In that regard, notice that if we let g denote the greatest common divisor of 



it follows that 


and 


where d_{k} is the denominator of the kth term in the above series for π/2. Now it appears that 



so g = [(k+1)!]^{2}, which implies that 



Equation (5) is a kind of convolution of the terms of the two sequences, which may be a more natural definition of their relation than the original linear 2nd order recurrence. 

Actually (5) is just one of a set of formulas relating the "crossproducts" of these two series. In general we have 



where Q(k) is a polynomial in k of degree j–1. The first few such polynomials are listed below 



The constant coefficients are just the values of the N sequence, whereas the leading coefficients 1, 2, 5, 12, 29, 70, ... are the "Pell numbers" that satisfy the recurrence p_{n} = 2p_{n–1} + p_{n–2}. Also, notice that the sums of the coefficients for the jth polynomial are 1, 5, 44, 476, 6336, 99504, ... which are the values of the other "basis sequence" mentioned previously. In other words, these values satisfy the same recurrence as do the N and D sequences, but with the initial values 1,0,... 

Incidentally, using the original sequence notation, we have the identity, applicable to any linear secondorder recurrence: 



This is discussed in the note on antisymmetric arrays for linear recurrences. 
