A295783 First differences of A293160.

0, 1, 1, 2, 2, 6, 7, 11, 17, 30, 40, 73, 109, 165, 269, 441, 675, 1076, 1671, 2699, 4256, 6726, 10585, 16969, 26524, 42386, 66979

Offset: 1

Author

R. J. Mathar, Nov 27 2017

Keywords

• nonn
• more

Comments

a(n) is the number of distinct numerators that exist in row n of the Kepler tree A294442 but not yet in row n-1 of the tree (assuming a row count such that 1/1 is in row 0).

It is the number of numerators that are "new" in row n (because the set of denominators of row n-1 contributes to the set of numerators of row n).

a(n) is nonnegative because A293160 is monotonically increasing (because all numerators of one row become numerators of the next row).

Define the "entry level" E(j) as the smallest row number at which denominator j appears in A294442 (again: row counts start at 1/1 as row 0), then a(n+1) is the number of occurrences of n in j: a(n+1) = #{j: E(j)=n}.

E(j) = A178047(j), as originally observed by R. J. Mathar, because every denominator j first appears both in Kepler's tree (used in E(j)) and in the left half of Stern-Brocot tree (used in A178047) when there is a fraction p/q with p+q=j in the previous row, and the rows of these two trees contain the same fractions (in different orders), assuming the row labeling from A178047 for Stern-Brocot tree. - Andrey Zabolotskiy, Dec 06 2024

Links

• R. J. Mathar, The Kepler tree of reduced fractions, (2017).

Crossrefs

Cf. A294442, A294443, A178047.

Programs

Mathematica

Differences@ Map[Length@ Union@ Numerator@ # &, #] &@ Nest[Append[#, Flatten@ Map[{#1/(#1 + #2), #2/(#1 + #2)} & @@ {Numerator@ #, Denominator@ #} &, Last@ #]] &, {{1/1}, {1/2}}, 21] (* Michael De Vlieger, Apr 18 2018 *)

Extensions

a(25)-a(27) from Michael De Vlieger, Apr 18 2018