OFFSET
2,1
COMMENTS
The base-n Champernowne constant (C_n) is normal in base n. A(n,k) is the (k+1)-th decimal digit of the fractional part of C_n.
LINKS
Verónica Becher and Santiago Figueira, An example of a computable absolutely normal number, Theoretical Computer Science, 270 (2002), 947-958.
Arthur H. Copeland and Paul Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857-860.
Davar Khoshnevisan, Normal Numbers are Normal, Clay Mathematics Institute Annual Report, 2006, 15-31.
Ivan Niven and H.S. Zuckerman, On The Definition of Normal Numbers, Pacific J. Math., 1 (1951), 103-109.
Davis Smith, A Sufficient Condition For Normalcy.
FORMULA
A(n,k) = floor(C_n*10^(k+1)) mod 10 where C_n (the base-n Champernowne constant) = Sum_{i>=1} i/(n^(i + Sum_{k=1..i-1} floor(log_m(k+1))).
EXAMPLE
The square array A(n,k) begins:
n/k | 0 1 2 3 4 5 6 7 8 9 10 11 ...
----+---------------------------------------
2 | 8 6 2 2 4 0 1 2 5 8 6 8 ...
3 | 5 9 8 9 5 8 1 6 7 5 3 8 ...
4 | 4 2 6 1 1 1 1 1 1 1 1 1 ...
5 | 3 1 0 7 3 6 1 1 1 1 1 1 ...
6 | 2 3 9 8 6 2 6 8 5 8 1 5 ...
7 | 1 9 4 4 3 5 5 3 5 0 8 6 ...
8 | 1 6 3 2 6 4 8 1 2 1 0 5 ...
9 | 1 4 0 6 2 4 9 7 6 1 1 9 ...
10 | 1 2 3 4 5 6 7 8 9 1 0 1 ...
...
MATHEMATICA
A[n_, k_]:=Mod[Floor[ChampernowneNumber[n]10^(k + 1)] , 10]; Flatten[Table[Reverse[Table[A[n-k, k], {k, 0, n-2}]], {n, 2, 14}]] (* Stefano Spezia, May 13 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Davis Smith, May 12 2022
STATUS
approved