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A295783 First differences of A293160.

%I A295783 #36 Dec 06 2024 11:22:12
%S A295783 0,1,1,2,2,6,7,11,17,30,40,73,109,165,269,441,675,1076,1671,2699,4256,
%T A295783 6726,10585,16969,26524,42386,66979
%N A295783 First differences of A293160.
%C A295783 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).
%C A295783 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).
%C A295783 a(n) is nonnegative because A293160 is monotonically increasing (because all numerators of one row become numerators of the next row).
%C A295783 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}.
%C A295783 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
%H A295783 R. J. Mathar, <a href="/A294443/a294443.pdf">The Kepler tree of reduced fractions</a>, (2017).
%t A295783 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 *)
%Y A295783 Cf. A294442, A294443, A178047.
%K A295783 nonn,more
%O A295783 1,4
%A A295783 _R. J. Mathar_, Nov 27 2017
%E A295783 a(25)-a(27) from _Michael De Vlieger_, Apr 18 2018