The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A071766 Denominator of the continued fraction expansion whose terms are the first-order differences of exponents in the binary representation of 4n, with the exponents of 2 being listed in descending order. 22
 1, 1, 1, 2, 1, 2, 3, 3, 1, 2, 3, 3, 4, 5, 4, 5, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 6, 9, 10, 11, 9, 12, 11, 13, 6, 9, 10, 11, 9, 12, 11, 13, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 6, 9, 10, 11, 9, 12, 11, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS If the terms (n>0) are written as an array: 1, 1, 2, 1, 2, 3, 3, 1, 2, 3, 3, 4, 5, 4, 5, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 6, 9, 10, 11, 9, 12, 11, 13, 6, 9, then the sum of the m-th row is 3^m (m = 0,1,2,3,...), each column is constant and the terms are from A071585 (a(2^m+k) = A071585(k), k = 0,1,2,...). If the rows are written in a right-aligned fashion:                                                                                  1,                                                                              1,  2,                                                                      1,  2,  3,  3,                                                        1, 2,  3,  3, 4,  5,  4,  5,                           1, 2,  3,  3, 4,  5,  4,  5, 5, 7,  7,  8, 5,  7,  7,  8, ..., 7, 7, 8, 5, 7, 7, 8, 6, 9, 10, 11, 9, 12, 11, 13, 6, 9, 10, 11, 9, 12, 11, 13, then each column is a Fibonacci sequence (a(2^(m+2)+k) = a(2^(m+1)+k) + a(2^m+k), m = 0,1,2,..., k = 0,1,2,...,2^m-1 with a_k(1) = A071766(k) and a_k(2) = A086593(k) being the first two terms of each column sequence). - Yosu Yurramendi, Jun 23 2014 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..10000 FORMULA a(n) = A071585(m), where m = n - floor(log_2(n)); a(0) = 1, a(2^k) = 1, a(2^k + 1) = 2. a(2^k - 1) = Fibonacci(k+1) = A000045(k+1). a(2^m+k) = A071585(k), m=0,1,2,..., k=0,1,2,...,2^m-1. - Yosu Yurramendi, Jun 23 2014 a(2^m-k) = F_k(m), k=0,1,2,..., m > log_2(k). F_k(m) is a Fibonacci sequence, where F_k(1) = a(2^(m_0(k))-1-k), F_k(2) = a(2^(m_0(k)+1)-1-k), m_0(k) = ceiling(log_2(k+1))+1 = A070941(k). - Yosu Yurramendi, Jun 23 2014 EXAMPLE a(37) = 5 as it is the denominator of 17/5 = 3 + 1/(2 + 1/2), which is a continued fraction that can be derived from the binary expansion of 4*37 = 2^7 + 2^4 + 2^2; the binary exponents are {7, 4, 2}, thus the differences of these exponents are {3, 2, 2}; giving the continued fraction expansion of 17/5 = [3,2,2]. 1, 2, 3, 3/2, 4, 5/2, 4/3, 5/3, 5, 7/2, 7/3, 8/3, 5/4, 7/5, 7/4, 8/5, 6, ... MATHEMATICA {1}~Join~Table[Denominator@ FromContinuedFraction@ Append[Abs@ Differences@ #, Last@ #] &@ Log2[NumberExpand[4 n, 2] /. 0 -> Nothing], {n, 120}] (* Version 11, or *) {1}~Join~Table[Denominator@ FromContinuedFraction@ Append[Abs@ Differences@ #, Last@ #] &@ Log2@ DeleteCases[# Reverse[2^Range[0, Length@ # - 1]] &@ IntegerDigits[4 n, 2], k_ /; k == 0], {n, 120}] (* Michael De Vlieger, Aug 15 2016 *) PROG (PARI) {a(n)=local(N=4*n, E=#binary(N)-1, P=[E], CF); while(E>1, P=concat(P, E=#binary(N=N-2^E)-1)); CF=Vec(Ser(P)*(x-1)); if(n==0, CF[1]=1, CF[1]=0); contfracpnqn(CF)[1, 1]} \\ Paul D. Hanna, Feb 22 2013 for(n=0, 256, print1(a(n), ", ")) (R) blocklevel <- 6 # arbitrary a <- 1 for(m in 0:blocklevel) for(k in 0:(2^(m-1)-1)){   a[2^(m+1)+k]             <- a[2^m+k]   a[2^(m+1)+2^(m-1)+k]     <- a[2^m+2^(m-1)+k]   a[2^(m+1)+2^m+k]         <- a[2^(m+1)+k]     +  a[2^(m+1)+2^(m-1)+k]   a[2^(m+1)+2^m+2^(m-1)+k] <- a[2^(m+1)+2^m+k] } a # Yosu Yurramendi, Jul 11 2014 CROSSREFS Cf. A071585, A086593. Sequence in context: A033803 A035531 A118977 * A007305 A112531 A100002 Adjacent sequences:  A071763 A071764 A071765 * A071767 A071768 A071769 KEYWORD easy,nonn,frac AUTHOR Paul D. Hanna, Jun 04 2002 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 28 09:16 EDT 2021. Contains 347714 sequences. (Running on oeis4.)