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%I #7 Nov 21 2017 15:26:36
%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,
%e 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; # A294443
%p dc; # A294444
%Y Cf. A294442, A294443.
%Y See A293160 for a similar sequence related to the Stern-Brocot triangle A002487.
%K nonn,more
%O 0,4
%A _N. J. A. Sloane_, Nov 20 2017
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