A177903 -id:A177903 - cp's OEIS frontend

A177407 Form triangle of weighted Farey fractions; read denominators by rows.

1, 1, 1, 3, 3, 1, 1, 5, 7, 3, 9, 9, 3, 7, 5, 1, 1, 7, 11, 5, 17, 19, 7, 17, 13, 3, 5, 7, 9, 27, 27, 9, 7, 5, 3, 13, 17, 7, 19, 17, 5, 11, 7, 1, 1, 9, 15, 7, 25, 29, 11, 27, 21, 5, 9, 13, 17, 53, 55, 19, 15, 11, 7, 31, 41, 17, 47, 43, 13, 29, 19, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9

Offset: 0

N. J. A. Sloane, Dec 10 2010

Start with the list of fractions 0/1, 1/1 and repeatedly insert the weighted mediants (2a+c)/(2b+d) and (a+2c)/(b+2d) between every pair of adjacent elements a/b and c/d of the list. The fractions are to be reduced before the insertion step.

			Triangle begins:
  0 1
  - -
  1 1
.
  0 1 2 1
  - - - -
  1 3 3 1
.
  0 1 2 1 4 5 2 5 4 1
  - - - - - - - - - -
  1 5 7 3 9 9 3 7 5 1

James Propp, Posting to the Math Fun Mailing List, Dec 10 2010.

Nathaniel Johnston, Table of n, a(n) for n = 0..29533 (first 10 rows of triangle)
Dhroova Aiylam and Tanya Khovanova, Weighted Mediants and Fractals, arXiv:1711.01475 [math.NT], 2017.

Cf. A177405, A177903, A006842/A006843.

a(44)-a(80) and some corrected terms from Nathaniel Johnston, Apr 12 2011

A177405 Form triangle of weighted Farey fractions; read numerators by rows.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 2, 3, 4, 13, 14, 5, 4, 3, 2, 9, 12, 5, 14, 13, 4, 9, 6, 1, 0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 2, 3, 4, 13, 14, 5, 4, 3, 2, 9, 12, 5, 14, 13, 4, 9, 6, 1, 4, 5, 2, 7, 8, 3, 10, 11

Offset: 0

N. J. A. Sloane, Dec 10 2010

James Propp asks: Does every fraction between 0 and 1 with odd denominator appear in the triangle?

			Triangle begins:
0 1
- -
1 1
0 1 2 1
- - - -
1 3 3 1
0 1 2 1 4 5 2 5 4 1
- - - - - - - - - -
1 5 7 3 9 9 3 7 5 1
0 1 .2 1 .4 .5 2 .5 .4 1 2 3 4 13 14 5 4 3 2 .9 12 5 14 13 4 .9 6 1
- - -- - -- -- - -- -- - - - - -- -- - - - - -- -- - -- -- - -- - -
1 7 11 5 17 19 7 17 13 3 5 7 9 27 27 9 7 5 3 13 17 7 19 17 5 11 7 1

James Propp, Posting to the Math Fun Mailing List, Dec 10 2010.

Nathaniel Johnston, Table of n, a(n) for n = 0..29533 (first 10 rows of triangle)
Dhroova Aiylam, Tanya Khovanova, Weighted Mediants and Fractals, arXiv:1711.01475 [math.NT], 2017.

Cf. A177407, A177903, A006842/A006843.

Mma code from James Propp:
        Lengthen[L_] :=
         Module[{i, M}, M = Table[0, {3 Length[L]}];
          M[[1]] = Numerator[L[[1]]]/(2 + Denominator[L[[1]]]);
          M[[2]] = 2*Numerator[L[[1]]]/(1 + 2 Denominator[L[[1]]]);
          For[i = 1, i < Length[L], i++, M[[3 i]] = L[[i]];
           M[[3 i + 1]] = (2 Numerator[L[[i]]] +
               Numerator[L[[i + 1]]])/(2 Denominator[L[[i]]] +
               Denominator[L[[i + 1]]]);
           M[[3 i + 2]] = (Numerator[L[[i]]] +
               2 Numerator[L[[i + 1]]])/(Denominator[L[[i]]] +
               2 Denominator[L[[i + 1]]])]; M[[3 Length[L]]] = L[[Length[L]]];
           Return[M]]
        WF[n_] := WF[n] = If[n == 0, {1}, Lengthen[WF[n - 1]]]

a(45)-a(80) and some corrected terms from Nathaniel Johnston, Apr 12 2011

A178031 Consider the Farey tree A049455/A049456; a(n) = row at which the denominator n first appears (assumes first row is labeled row 1).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 9, 9, 8, 9, 9, 9, 10, 9, 9, 9, 10, 9, 9, 9, 9, 9, 10, 9, 9, 10, 10, 10, 11, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, Dec 16 2010

nonn

Computed by Alan Wechsler, Dec 16 2010.
Richard C. Schroeppel also asked about the analogous sequence giving the last occurrence of denominator n.
The first occurrence of k in this sequence is apparently at n = A135510(k-1), except for k=5. The last occurrence of k is at n = Fibonacci(k). - Andrey Zabolotskiy, Dec 01 2024

			Start with a pair of fractions 0/1, 1/1 and repeatedly insert the "Farey sum" (p+r)/(q+s) in between every pair of adjacent fractions p/q, r/s. The first few iterations are:
1:   0/1                                     1/1
2:   0/1                 1/2                 1/1
3:   0/1       1/3       1/2       2/3       1/1
4:   0/1  1/4  1/3  2/5  1/2  3/5  2/3  3/4  1/1
We only look at the denominators in this table (which form the sequence A049456, or A002487 if the rightmost column is removed).
1 first appears in row 1, so a(1) = 1.
2 first appears in row 2, so a(2) = 2.
3 first appears in row 3, so a(3) = 3.
4 and 5 first appear in row 4, so a(4) = a(5) = 4.

Based on a posting by Richard C. Schroeppel to the Math Fun Mailing List, Dec 15 2010.

Bo Gyu Jeong, Table of n, a(n) for n = 1..10000
Richard J. Mathar, The Kepler binary tree of reduced fractions, 2017.

See A178047 for another version. Cf. A002487, A006842, A006843, A177903, A178042, A135510.

More terms from Bo Gyu Jeong, Oct 20 2012

A178042 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 1).

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 5, 4, 4, 5, 5, 5, 4, 5, 5, 6, 6, 6, 6, 5, 5, 6, 5, 6, 7, 5, 5, 7, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 7, 8, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 6, 7, 8, 7, 7, 7, 7, 7, 7, 7, 6, 7, 6, 8, 8, 7, 7, 8, 8, 7, 8, 7, 7, 7, 6, 6, 8, 7, 7, 7, 8, 8, 8, 7, 7, 7, 8, 8, 7, 8, 8, 8, 7, 8, 8

Offset: 0

N. J. A. Sloane, Dec 16 2010

nonn

Equals A177903 + 1. See that entry for further information.

cp's OEIS Frontend

A177407 Form triangle of weighted Farey fractions; read denominators by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A177405 Form triangle of weighted Farey fractions; read numerators by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A178031 Consider the Farey tree A049455/A049456; a(n) = row at which the denominator n first appears (assumes first row is labeled row 1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A178042 Consider the weighted Farey tree A177405/A177407; a(n) = row at which the denominator 2n+1 first appears (assumes first row is labeled row 1).

Views

Author

Keywords

Comments