A086593 Bisection of A086592, denominators of the left-hand half of Kepler's tree of fractions.

2, 3, 4, 5, 5, 7, 7, 8, 6, 9, 10, 11, 9, 12, 11, 13, 7, 11, 13, 14, 13, 17, 15, 18, 11, 16, 17, 19, 14, 19, 18, 21, 8, 13, 16, 17, 17, 22, 19, 23, 16, 23, 24, 27, 19, 26, 25, 29, 13, 20, 23, 25, 22, 29, 26, 31, 17, 25, 27, 30, 23, 31, 29, 34, 9, 15, 19, 20, 21, 27, 23, 28, 21, 30

Offset: 1

Author

Antti Karttunen, Aug 28 2003

Keywords

• nonn

Comments

Also denominator of alternate fractions in Kepler's tree as shown in A294442. - N. J. A. Sloane, Nov 20 2017

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2048 (computed from b-file of A020650 provided by _T. D. Noe_)

Crossrefs

Cf. A000045, A020650, A020651, A071585, A071766, A086592, A294442.

Programs

Mathematica

(* b = A020650 *) b[1] = 1; b[2] = 2; b[3] = 1; b[n_] := b[n] = Switch[ Mod[n, 4], 0, b[n/2 + 1] + b[n/2], 1, b[(n - 1)/2 + 1], 2, b[(n - 2)/2 + 1] + b[(n - 2)/2], 3, b[(n - 3)/2]]; a[1] = 2; a[n_] := b[4 n - 4]; Array[a, 100] (* Jean-François Alcover, Jan 22 2016, after Yosu Yurramendi's formula for A020650 *)

R

maxlevel <- 15
d <- c(1,2)
for(m in 0:maxlevel)
 for(k in 1:2^m) {
   d[2^(m+1)    +k] <- d[k] + d[2^m+k]
   d[2^(m+1)+2^m+k] <- d[2^(m+1)+k]
}
a <- vector()
for(m in 0:maxlevel) for(k in 0:(2^m-1)) a[2^m+k] <- d[2^(m+1)+k]
  a[1:63]
# Yosu Yurramendi, May 16 2018

Formula

a(n) = A086592(2n-1) = A020650(4n-2).

a(n+1) = A071585(n) + A071766(n), n >= 0. - Yosu Yurramendi, Jun 30 2014

From Yosu Yurramendi, Jan 04 2016: (Start)

a(2^(m+1)+k+1) - a(2^m+k+1) = A071585(k), m >= 0, 0 <= k < 2^m.

a(2^(m+2)-k) = a(2^(m+1)-k) + a(2^m-k), m > 0, 0 <= k < 2^m-1.

(End)

a(2^n) = A000045(n+3). - Antti Karttunen, Jan 29 2016, based on above.

a(n) = A020651(4n-1), a(n+1) = A020651(4n+1), n > 0. - Yosu Yurramendi, May 08 2018

a(2^m+k) = A071585(2^(m+1)+k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, May 16 2018