/* The function fusc is related to Calkin-Wilf sequence */
fusc(n):=block(
    [k:n,a:1,b:0],
    while k>0 do (if evenp(k) then (k:k/2,a:a+b) else (k:(k-1)/2,b:a+b)),
    b)$

/* Calkin-Wilf function using fusc */
calkin_wilf(n):=fusc(n)/fusc(n+1)$

/* Function that given a nonnegative rational returns its position in the Calkin-Wilf sequence */
cf_bin(fracti):=block(
    cf_list:cf(fracti),
    cf_len:length(cf_list),
    if oddp(cf_len) then cf_list:reverse(cf_list) else cf_list:reverse(append(at(cf_list,[cf_list[cf_len]=cf_list[cf_len]-1]),[1])),
    makelist(lambda([x],if oddp(x) then makelist(1,j,1,cf_list[x]) else makelist(0,j,1,cf_list[x]))(i),i,1,length(cf_list)), /* decoding part begins here */
    apply(append,%%),
    apply("+",makelist(2^i,i,0,length(%%)-1)*reverse(%%)))$

/* Test cases */
/* 20 first terms of the sequence */
makelist(calkin_wilf(i),i,1,20);

/* Position of 83116/51639 in Calkin-Wilf sequence */
83116/51639$
cf_bin(%);