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%I #18 Jan 13 2024 10:41:56
%S 0,1,3,2,7,6,4,5,14,15,13,12,8,9,10,11,28,29,30,31,27,26,24,25,16,17,
%T 18,19,20,21,22,23,56,57,58,59,60,61,62,63,54,55,53,52,48,49,50,51,32,
%U 33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,112,113,114,115,116,117
%N Permutation of nonnegative integers induced by wreath recursion a=s(b,c), b=s(c,a), c=(c,c), starting from state a, rewriting bits from the second most significant bit toward the least significant end.
%C This permutation of natural numbers is induced by the first generator of group 2861 mentioned on page 144 of "Classification of groups generated by 3-state automata over a 2-letter alphabet" paper. It can be computed by starting scanning n's binary expansion rightward from the second most significant bit, complementing every bit down to and including A) either the first 0-bit at even distance from the most significant bit or B) the first 1-bit at odd distance from the most significant bit.
%H Antti Karttunen, <a href="/A154448/b154448.txt">Table of n, a(n) for n = 0..2047</a>
%H Bondarenko, Grigorchuk, Kravchenko, Muntyan, Nekrashevych, Savchuk, and Sunic, <a href="http://arxiv.org/abs/0803.3555">Classification of groups generated by 3-state automata over a 2-letter alphabet</a>, arXiv:0803.3555 [math.GR], 2008, p. 144.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e 25 = 11001 in binary, the first zero-bit at odd distance from the msb is immediately at where we start (at the second most significant bit), so we complement it and fix the rest, yielding 10001 (17 in binary), thus a(25)=17.
%o (MIT/GNU Scheme) (define (A154448 n) (if (< n 2) n (let loop ((maskbit (A072376 n)) (p 1) (z n)) (cond ((zero? maskbit) z) ((= p (modulo (floor->exact (/ n maskbit)) 2)) (+ z (* (- 1 (* 2 p)) maskbit))) (else (loop (floor->exact (/ maskbit 2)) (- 1 p) (- z (* (- 1 (* 2 p)) maskbit))))))))
%o (R)
%o maxlevel <- 5 # by choice
%o a <- 1
%o for(m in 0:maxlevel) {
%o for(k in 0:(2^m-1)){
%o a[2^(m+1) + 2*k ] <- 2*a[2^m + k]
%o a[2^(m+1) + 2*k + 1] <- 2*a[2^m + k] + 1
%o }
%o x <- floor(2^(m+2)/3)
%o a[2*x ] <- 2*a[x] + 1
%o a[2*x + 1] <- 2*a[x]
%o }
%o (a <- c(0, a))
%o # _Yosu Yurramendi_, Oct 12 2020
%Y Inverse: A154447. a(n) = A054429(A154447(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154446. Corresponds to A154458 in the group of Catalan bijections.
%K nonn,base
%O 0,3
%A _Antti Karttunen_, Jan 17 2009
%E Spelling/notation corrections by _Charles R Greathouse IV_, Mar 18 2010
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