# [seqfan] Re: Is 0001 a 4-digit base 10 number ?

israel at math.ubc.ca israel at math.ubc.ca
Fri Aug 19 20:51:26 CEST 2011

```Indeed, for the base-b version, let M be the b x b tridiagonal matrix with
1's on the main diagonal and the diagonals above and below it, 0's
everywhere else, and e the b-dimensional column vector of all 1's, then
a(n) = e^T M^(n-1) e for n >= 1, and the generating function is g(z) = 1 +
z e^T (I - z M)^{-1} e, which is a rational function with poles at the
reciprocals of the eigenvalues of M. A linear recurrence for n >= 1 is
sum_{j=0}^b c_j a(n+j) = 0 where the characteristic polynomial of M is
sum_{j=0}^b c_j t^j. In particular for b=10 the recurrence is
-a[n]+4*a[n+1]+18*a[n+2]-36*a[n+3]-35*a[n+4]+84*a[n+5]-14*a[n+6]-48*a[n+7]
+36*a[n+8]-10*a[n+9]+a[n+10] = 0

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada

On Aug 19 2011, franktaw at netscape.net wrote:

>Incidently, A126364 has a rational generating function (and
>equivalently a linear recurrence).
>
>Note that the base 3 equivalent is A078057 (essentially the same as
>A001333).
>