Digit sums of binomial sums

For odd n>=3, we consider a general hypothetical identity for the differences S_{n,0}(x) of multiples of n with even and odd digit sums in the base n-1 in interval [0,x), which we prove in the cases n=3 and n=5 and empirically confirm for some other n. We give a verification algorithm for this identity for any odd n. The hypothetical identity allows to give a general recursion for S_{n,0}(x) for every integer x depending on the residue of x modulo p(n)=2n(n-1)^{n-1}, such that p(3)=24, p(5)=2560, p(7)=653184, etc.

Fibonacci Quarterly, 1991

2013

Given integers a,b>1 with _ba irrational, we investigate the connection between the conjectured asymptotic equidistribution of digits in the base-b expansion of a^n and the (non-)Diophantine properties of the number θ_a,b=_ba.

Journal of Number Theory, 1997

We will show that the sum-of-digits function has an asymptotic Gaussian behaviour, and we deduce some new summational formulae. More precisely we consider numeration systems relative to substitutive bases, whose general case contains among others the case of Parry linear recurrent bases (as the Fibonacci sequence for instance).

Journal of Number Theory, 1990

Extensions and improvements of a recent paper, "On Digit Expansions with Respect to Linear Recurrences" by A. Petho and R. F. Tichy (J. Number Theory 33 (1989), 243-256) are established. Furthermore distribution properties mod 1 of the sequence @a(n)) are investigated, where so(n) denotes the sum-of-digits function with respect to the linear recurrence G. a

Proceedings of the Edinburgh Mathematical …, 2007

Journal de Théorie des Nombres de Bordeaux, 2004

The Ramanujan Journal, 2005

Journal of Number Theory, 1989

An explicit formula for the mean value of the sum-of-digits function with respect to linear recurring sequences is established. Thus a recent paper of J. Coquet and P. Van Den Bosch on the Fibonacci number system is extended to the general case. Q

Journal of Mathematical Analysis and Applications, 2011

Let φ(x) = 2 inf{|x − n| : n ∈ Z}, and define for α > 0 the function f α (x) = ∞ j=0 1 2 αj φ(2 j x). Tabor and Tabor [J. Math. Anal. Appl. 356 (2009), 729-737] recently proved the inequality f α x + y 2 ≤ f α (x) + f α (y) 2 + |x − y| α , for α ∈ [1, 2]. By developing an explicit expression for f α at dyadic rational points, it is shown in this paper that the above inequality can be reduced to a simple inequality for weighted sums of binary digits. That inequality, which seems of independent interest, is used to give an alternative proof of the result of Tabor and Tabor, which captures the essential structure of f α .