A294443 Number of distinct numbers appearing as numerators in row n of Kepler's triangle A294442.

1, 1, 2, 3, 5, 7, 13, 20, 31, 48, 78, 118, 191, 300, 465, 734, 1175, 1850, 2926, 4597, 7296, 11552, 18278, 28863, 45832, 72356, 114742, 181721

Offset

0,3

Comments

It would be nice to have a formula or recurrence.

Links

Table of n, a(n) for n=0..27.

R. J. Mathar, Java program to compute the sequence

R. J. Mathar, The Kepler binary tree of reduced fractions

Example

Row 4 of A294442 contains eight fractions: 1/5, 4/5, 3/7, 4/7, 2/7, 2/7, 3/8, 5/8.
There are five distinct numerators, so a(4) = 5.

Maple

# 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
S[0]:=[[1, 1]]; S[1]:=[[1, 2]];
nc:=[1, 1]; dc:=[1, 1];
for n from 2 to 18 do
S[n]:=[];
for k from 1 to nops(S[n-1]) do
t1:=S[n-1][k];
a:=[t1[1], t1[1]+t1[2]];
b:=[t1[2], t1[1]+t1[2]];
S[n]:=[op(S[n]), a, b];
od:
listn:={};
for k from 1 to nops(S[n]) do listn:={op(listn), S[n][k][1]}; od:
c:=nops(listn);  nc:=[op(nc), c];
listd:={};
for k from 1 to nops(S[n]) do listd:={op(listd), S[n][k][2]}; od:
c:=nops(listd);  dc:=[op(dc), c];
od:
nc; # A294443
dc; # A294444

Mathematica

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, A294444, A295783 (first differences).

See A293160 for a similar sequence related to the Stern-Brocot triangle A002487.

Sequence in context: A104189 A301776 A178570 * A293160 A249309 A250253

Adjacent sequences: A294440 A294441 A294442 * A294444 A294445 A294446

Keyword

nonn,more

Author

N. J. A. Sloane, Nov 20 2017

Extensions

a(22)-a(28) from Michael De Vlieger, Apr 18 2018

Status

approved