A165025 Number of cycles of n-digit numbers (including fixed points) under the base-4 Kaprekar map A165012.

1, 0, 1, 2, 1, 3, 1, 4, 3, 5, 4, 8, 5, 10, 8, 12, 10, 16, 12, 19, 16, 22, 19, 27, 22, 31, 27, 35, 31, 41, 35, 46, 41, 51, 46, 58, 51, 64, 58, 70, 64, 78, 70, 85, 78, 92, 85, 101, 92, 109, 101, 117, 109, 127, 117, 136, 127, 145, 136, 156, 145, 166, 156, 176, 166, 188, 176

Offset: 1

Author

Joseph Myers, Sep 04 2009

Keywords

• base
• nonn

Links

• Joseph Myers, Table of n, a(n) for n=1..200
• H. Hanslik, E. Hetmaniok, I. Sobstyl, et al., Orbits of the Kaprekar's transformations-some introductory facts, Zeszyty Naukowe Politechniki Śląskiej, Seria: Matematyka Stosowana z. 5, Nr kol. 1945; 2015.
• Index entries for the Kaprekar map

Crossrefs

Cf. A165012, A165017, A165026, A165027.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165006 (base 3), A165045 (base 5), A165064 (base 6), A165084 (base 7), A165103 (base 8), A165123 (base 9), A164731 (base 10).

Formula

Conjectures from Colin Barker, Jun 01 2017: (Start)

G.f.: x*(1 - x^2 + x^3 - 2*x^6 + 3*x^8 - x^10) / ((1 - x)^3*(1 + x)^2*(1 + x + x^2)).

a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n>7.

(End)