DigitCount

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

{(Subscript[s, b]^((1)))(n),(Subscript[s, b]^((2)))(n),…,(Subscript[s, b]^((b-1)))(n),(Subscript[s, b]^((0)))(n)} is a nonanalytical vector-valued function of n and b which is defined for integer n and positive integers b (b>1).

Symmetries and periodicities

Parity

{(Subscript[s, b]^((1)))(n),(Subscript[s, b]^((2)))(n),…,(Subscript[s, b]^((b-1)))(n),(Subscript[s, b]^((0)))(n)} is an even function with respect to n.

Periodicity

No periodicity

Series representations

Other series representations

Identities

Functional identities

Summation

Finite summation

Infinite summation

Products

Infinite products

Inequalities

Other identities

Congruence properties

Theorems

Numbers in Pascal's triangle

The number of odd numbers in the n th row of Pascal's triangle is given by (2^((Subscript[s, 2]^((1)))(n))).

The classical Thue-Morse sequence

The sequence (((-1))^((Subscript[s, 2]^((1)))(n))) is the classical Thue–Morse sequence, which does not contain any triple repetition of a subsequence.

History

– E. Waring (1770)
– E. Kummer (1852)
– E. Lucas (1878)
– M.D. Ocagne (1886)
– L.E. Dickson (1896)
– J. W. L. Glaisher (1899)
– S. Ramanujan
– J. R. Trollope (1968)
– D. R. Woods (1978)
– D. Robbins (1979)

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