A294444 - OEIS

%I #13 Dec 06 2024 11:11:07

%S 1,1,1,2,3,6,10,16,29,51,83,148,246,402,650,1084,1740,2803,4458

%N Number of distinct numbers appearing as denominators in row n of Kepler's triangle A294442.

%C It would be nice to have a formula or recurrence.

%e Row 4 of A294442 contains eight fractions, 1/5, 4/5, 3/7, 4/7, 2/7, 2/7, 3/8, 5/8.

%e There are three distinct denominators, so a(4) = 3.

%p # S[n] is the list of fractions, written as pairs [i, j], in row n of Kepler's triangle; nc is the number of distinct numerators, and dc the number of distinct denominators

%p S[0]:=[[1,1]]; S[1]:=[[1,2]];

%p nc:=[1,1]; dc:=[1,1];

%p for n from 2 to 18 do

%p S[n]:=[];

%p for k from 1 to nops(S[n-1]) do

%p t1:=S[n-1][k];

%p a:=[t1[1],t1[1]+t1[2]];

%p b:=[t1[2],t1[1]+t1[2]];

%p S[n]:=[op(S[n]),a,b];

%p od:

%p listn:={};

%p for k from 1 to nops(S[n]) do listn:={op(listn), S[n][k][1]}; od:

%p c:=nops(listn);  nc:=[op(nc),c];

%p listd:={};

%p for k from 1 to nops(S[n]) do listd:={op(listd), S[n][k][2]}; od:

%p c:=nops(listd);  dc:=[op(dc),c];

%p od:

%p nc; # A293160

%p dc; # this sequence

%Y Cf. A294442.

%Y See A293160 for the number of distinct numerators, or for numerators or denominators in the Stern-Brocot triangle A002487.

%K nonn,more

%O 0,4

%A _N. J. A. Sloane_, Nov 20 2017