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

1,1

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)

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A265535 A094802 A075084 * A073757 A088468 A355729

Adjacent sequences: A086590 A086591 A086592 * A086594 A086595 A086596

Keyword

nonn

Author

Antti Karttunen, Aug 28 2003

Status

approved