A295783 First differences of A294443.

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

1,4

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 A294443 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}.

It is likely that E(j) = A178047(j), verified for the first 10000 terms.

Links

Table of n, a(n) for n=1..27.

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

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 *)

Crossrefs

Cf. A294442, A294443, A178047.

Sequence in context: A079811 A097869 A298079 * A060303 A099577 A268500

Adjacent sequences: A295780 A295781 A295782 * A295784 A295785 A295786

Keyword

nonn,more

Author

R. J. Mathar, Nov 27 2017

Extensions

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

Status

approved