A fancy way to obtain the binary digits of $759250125\sqrt{2}$

TL;DR

The paper introduces an extension of a known sequence that enables a novel method to extract the binary digits of multiples of , including specific large numbers like 759250125, in a mathematically elegant way.

Contribution

It presents a new sequence-based technique to obtain binary digits of scaled , extending previous work to larger and more complex numbers.

Findings

Successfully generalizes Graham--Pollak's sequence for larger numbers

Demonstrates extraction of binary digits of 759250125

Provides a mathematical framework for digit extraction of scaled 

Abstract

R. L. Graham and H. O. Pollak observed that the sequence $$u_1=1,\qquad u_{n+1}=\lfloor \sqrt{2} (u_n+1/2)\rfloor, \quad n\geq 1,$$ has the curious property that the sequence of numbers $(u_{2n+1}-2u_{2n-1})_{n\geq 1}$ denotes the binary digits of $\sqrt{2}$. We present an extension of Graham--Pollak's sequence which allows to get -- in a fancy way -- the binary digits of $759250125\sqrt{2}$ and other numbers.