A086593 - OEIS

%I #46 Jul 14 2018 17:06:03

%S 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,

%T 19,14,19,18,21,8,13,16,17,17,22,19,23,16,23,24,27,19,26,25,29,13,20,

%U 23,25,22,29,26,31,17,25,27,30,23,31,29,34,9,15,19,20,21,27,23,28,21,30

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

%C Also denominator of alternate fractions in Kepler's tree as shown in A294442. - _N. J. A. Sloane_, Nov 20 2017

%H Antti Karttunen, <a href="/A086593/b086593.txt">Table of n, a(n) for n = 1..2048</a> (computed from b-file of A020650 provided by _T. D. Noe_)

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

%F a(n+1) = A071585(n) + A071766(n), n >= 0. - _Yosu Yurramendi_, Jun 30 2014

%F From _Yosu Yurramendi_, Jan 04 2016: (Start)

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

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

%F (End)

%F a(2^n) = A000045(n+3). - _Antti Karttunen_, Jan 29 2016, based on above.

%F a(n) = A020651(4n-1), a(n+1) = A020651(4n+1), n > 0. - _Yosu Yurramendi_, May 08 2018

%F a(2^m+k) = A071585(2^(m+1)+k),  m >= 0, 0 <= k < 2^m. - _Yosu Yurramendi_, May 16 2018

%t (* 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 *)

%o (R)

%o   maxlevel <- 15

%o d <- c(1,2)

%o for(m in 0:maxlevel)

%o  for(k in 1:2^m) {

%o    d[2^(m+1)    +k] <- d[k] + d[2^m+k]

%o    d[2^(m+1)+2^m+k] <- d[2^(m+1)+k]

%o }

%o a <- vector()

%o for(m in 0:maxlevel) for(k in 0:(2^m-1)) a[2^m+k] <- d[2^(m+1)+k]

%o   a[1:63]

%o # _Yosu Yurramendi_, May 16 2018

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

%K nonn

%O 1,1

%A _Antti Karttunen_, Aug 28 2003