OFFSET
0,1
COMMENTS
Number of collisions occurring in a system consisting of an infinitely massive, rigid wall at the origin, a ball with mass m stationary at position x1 > 0, and a ball with mass (10^2n)m at position x2 > x1 and rolling toward the origin, assuming perfectly elastic collisions and no friction. - Richard Holmes, Jun 17 2021 [Strictly speaking, this property, which is equivalent to the statement that the interval (m*Pi, Pi/arctan(1/m)) does not contain an integer for all m = 10^n, is not known to be true for sure. In other words, we do not know for certain that A332045 does not contain a power of 10. This is mentioned in the 2025 3Blue1Brown video "Why colliding blocks compute pi" which is a follow-up of the 2019 video. - Jianing Song, Sep 18 2025]
Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than 10^-9. - Paolo Xausa, Jul 15 2023
REFERENCES
Martin Gardner, Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American Magazine, W. H. Freemand and Company, New York, NY, 1992, pp. 274-275.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..100
G. Galperin, Playing pool with π (the number π from a billiard point of view), Regular and Chaotic Dynamics, 8 (2003), 375-394.
Wolfgang Haken, An attempt to understand the four color problem, in Journal of Graph Theory, Vol. 1, Issue 3, 1977, pp. 193-206.
G. Sanderson, Why do colliding blocks compute pi?, a 3Blue1Brown YouTube video, Jan 20 2019.
Grant Sanderson, Why colliding blocks compute pi, 3Blue1Brown video (2025).
FORMULA
a(n) = floor(Pi*10^n).
MATHEMATICA
s=RealDigits[Pi, 10, 30][[1]]; Table[FromDigits[Take[s, n]], {n, Length[s]}]
(* Alternative: *)
a[n_] := IntegerPart[Pi*10^n]; Table[a[n], {n, 0, 9}] (* Peter Luschny, Mar 15 2024 *)
PROG
(PARI) A011545(n)={localprec(n+3); Pi\10^-n} \\ M. F. Hasler, Mar 15 2024
CROSSREFS
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
Definition corrected by M. F. Hasler, Mar 15 2024
STATUS
approved
