A353962 Square array read by descending antidiagonals: The n-th row gives the decimal expansion of the base-n Champernowne constant.
8, 6, 5, 2, 9, 4, 2, 8, 2, 3, 4, 9, 6, 1, 2, 0, 5, 1, 0, 3, 1, 1, 8, 1, 7, 9, 9, 1, 2, 1, 1, 3, 8, 4, 6, 1, 5, 6, 1, 6, 6, 4, 3, 4, 1, 8, 7, 1, 1, 2, 3, 2, 0, 2, 1, 6, 5, 1, 1, 6, 5, 6, 6, 3, 0, 0, 8, 3, 1, 1, 8, 5, 4, 2, 4, 9, 9, 0, 0, 8, 1, 1, 5, 3, 8, 4, 5, 9, 9, 9, 0
Offset: 2
Examples
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 ... ...
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.
Programs
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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 *)
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_n(k+1)))).
Comments