A119608 Let b(1)=0, b(2)= 1. b(2^m +k) = (b(2^m+1-k) + b(k))/2, 1 <= k <= 2^m, m >= 0. a(n) is numerator of b(n).

0, 1, 1, 1, 1, 3, 3, 1, 1, 7, 5, 3, 3, 5, 7, 1, 1, 15, 9, 7, 5, 11, 13, 3, 3, 13, 11, 5, 7, 9, 15, 1, 1, 31, 17, 15, 9, 23, 25, 7, 5, 27, 21, 11, 13, 19, 29, 3, 3, 29, 19, 13, 11, 21, 27, 5, 7, 25, 23, 9, 15, 17, 31, 1, 1, 63, 33, 31, 17, 47, 49, 15, 9, 55, 41, 23, 25, 39, 57, 7, 5, 59, 37

Offset: 1

Author

Leroy Quet, Jun 04 2006

Keywords

• easy
• nonn
• frac

Comments

Denominator of b(n), for n >= 2, is A053644(n-1).

Links

• Nicholas Hillier, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A053644.

Programs

Maple

A119608 := proc (mmax) local a,b,m,k,bn,i; b := [0,1] ; for m from 1 to mmax do for k from 1 to 2^m do bn := (b[2^m+1-k]+b[k])/2 ; b := [op(b),bn] ; od ; od ; a := [] ; for i from 1 to nops(b) do a := [op(a),numer(b[i])] ; od ; RETURN(a) ; end: an := A119608(7) : for i from 1 to nops(an) do printf("%d,",an[i]) ; od ; # R. J. Mathar, Aug 06 2006

R

maxlevel <- 8 # by choice
b <- c(0,1)
for(m in 1:maxlevel) for(k in 1:2^m) b[2^m +k] = (b[2^m+1-k] + b[k])/2
d <- vector()
for(m in 0:maxlevel) for(k in 0:(2^m-1)) d[2^m + k] <- 2^m; d <- c(0,d)
a <- b*d
a[1:100]
# Yosu Yurramendi, Feb 05 2019

R

a <- 1
maxlevel <- 15 # by choice
for(m in 1:5) {
  a[2^(m+1)-1] <- 1
  a[2^(m+1)  ] <- 1
  for(k in 1:(2^m-1)){
  a[2^(m+1)-1-k] <- a[2^m+k]
  a[2^(m+1)  +k] <- a[2^m+k]+2^(m-floor(log2(k)))*a[k]
}}
a <- c(0,a)
a[1:128]
# Yosu Yurramendi, Mar 13 2019

Formula

From Yosu Yurramendi, Mar 13 2019: (Start)

Without a(1) = 0, and shifting the terms one place left:

a(2^m) = 1, m >= 0;

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

a(2^(m+1)+k) = a(2^m+k)+2^(m-floor(log_2(k)))*a(k), m >= 0, 0 < k < 2^m.

(End)

Extensions

More terms from R. J. Mathar, Aug 06 2006