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

2,1

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

Table of n, a(n) for n=2..92.

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

Rows: A066716 (n=2), A077771 (n=3), A033307 (n=10).

Cf. A063945.

Sequence in context: A011329 A092033 A061360 * A120818 A125579 A202258

Adjacent sequences: A353959 A353960 A353961 * A353963 A353964 A353965

Keyword

nonn,base,tabl

Author

Davis Smith, May 12 2022

Status

approved