cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A057432 Obtained by reading first the numerator then the denominator of fractions in left-hand half of Stern-Brocot tree (A007305/A007306).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 2, 5, 3, 5, 3, 4, 1, 5, 2, 7, 3, 8, 3, 7, 4, 7, 5, 8, 5, 7, 4, 5, 1, 6, 2, 9, 3, 11, 3, 10, 4, 11, 5, 13, 5, 12, 4, 9, 5, 9, 7, 12, 8, 13, 7, 11, 7, 10, 8, 11, 7, 9, 5, 6, 1, 7, 2, 11, 3, 14, 3, 13, 4, 15, 5, 18, 5, 17, 4, 13, 5, 14, 7, 19, 8, 21, 7, 18, 7, 17, 8, 19, 7
Offset: 0

Views

Author

N. J. A. Sloane, Sep 08 2000

Keywords

Examples

			The tree begins:
                                     1/1
                                     1/2
                  1/3                                   2/3
        1/4                 2/5               3/5                 3/4
    1/5      2/7       3/8       3/7     4/7       5/8       5/7      4/5
  1/6 2/9 3/11 3/10 4/11 5/13 5/12 4/9 5/9 7/12 8/13 7/11 7/10 8/11 7/9 5/6

Links

Crossrefs

Cf. A007305, A047679, A007306, A002487, A057431.
Related to the Kepler tree A294442 via row permutations given by A088208 or A131271.

Programs

  • Mathematica
    sbt[n_]:=Module[{P,L,Y},P={{1,0},{1,1}};L={{1,1},{0,1}};Y={{1,0},{0,1}}; w[b_]:=Fold[ #1.If[ #2==0,L,P]&,Y,b]; u[a_]:={a[[2,1]]+a[[2,2]],a[[1,1]]+a[[1,2]]}; s[l_]:={l,{Last[l],First[l]}}; Map[s,Map[u,Map[w,Part[Partition[Tuples[{0,1},n],2^(n-1)],1]]]]]
Flatten[Append[{1,1},Table[Map[First,sbt[i]],{i,1,6}]]] (* Peter Luschny, Apr 27 2009 *)

Extensions

More terms from Alford Arnold, Sep 11 2000
More terms from Joshua Zucker, May 11 2006

A294446 The tree of Farey fractions (or the Stern-Brocot tree), read across rows (the fraction i/j is represented as the pair i,j).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 0, 1, 1, 3, 1, 2, 2, 3, 1, 1, 0, 1, 1, 4, 1, 3, 2, 5, 1, 2, 3, 5, 2, 3, 3, 4, 1, 1, 0, 1, 1, 5, 1, 4, 2, 7, 1, 3, 3, 8, 2, 5, 3, 7, 1, 2, 4, 7, 3, 5, 5, 8, 2, 3, 5, 7, 3, 4, 4, 5, 1, 1, 0, 1, 1, 6, 1, 5, 2, 9, 1, 4, 3, 11, 2, 7, 3, 10, 1, 3, 4, 11, 3, 8, 5, 13, 2, 5, 5, 12, 3, 7, 4, 9, 1, 2, 5, 9, 4
Offset: 0

Views

Author

N. J. A. Sloane, Nov 21 2017

Keywords

Comments

The first row contains the fractions 0/1, 1/1,
and thereafter we copy the previous row, interpolating (a+c)/(b+d) between each pair of adjacent fractions a/b, c/d.
This version of the Farey tree contains the fractions in the range [0,1].
If we just look at the numerators we get A049455 and if we just look at the denominators we get A086596.

Examples

			This version of the tree begins as follows:
.................0/1..1/1
...............0/1..1/2..1/1
..........0/1..1/3..1/2..2/3..1/1
0/1..1/4..1/3..2/5..1/2..3/5..2/3..3/4..1/1
...
With the fractions written as pairs, the first few rows are:
[[0, 1], [1, 1]],
[[0, 1], [1, 2], [1, 1]],
[[0, 1], [1, 3], [1, 2], [2, 3], [1, 1]],
[[0, 1], [1, 4], [1, 3], [2, 5], [1, 2], [3, 5], [2, 3], [3, 4], [1, 1]],
[[0, 1], [1, 5], [1, 4], [2, 7], [1, 3], [3, 8], [2, 5], [3, 7], [1, 2], [4, 7,], [3, 5], [5, 8], [2, 3], [5, 7], [3, 4], [4, 5], [1, 1]]
...

References

  • W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
  • See A007305, A007306, A049455, A049456, etc. for many other references and links about the tree of Farey fractions (of which there are many versions).

Links

Crossrefs

Cf. A007305, A007306, A049455, A049456.
See A294442 for Kepler's tree of fractions.
For the number of distinct numerators in row n, see A293165, and for the distinct denominators see A293160.

Programs

  • Maple
    # S[n] is the list of fractions, written as pairs [i, j], in row n of the triangle of Farey fractions
S[0]:=[[0, 1], [1, 1]];
for n from 1 to 6 do
S[n]:=[[0,1]];
for k from 1 to nops(S[n-1])-1 do
a:=S[n-1][k][1]+S[n-1][k+1][1];
b:=S[n-1][k][2]+S[n-1][k+1][2];
S[n]:=[op(S[n]), [a, b], S[n-1][k+1]];
od:
lprint(S[n]);
od:

A295783 First differences of A293160.

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Nov 27 2017

Keywords

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 A293160 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}.
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

Links

Crossrefs

Cf. A294442, A294443, A178047.

Programs

  • 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 *)

Extensions

a(25)-a(27) from Michael De Vlieger, Apr 18 2018
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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