A267111 - OEIS

%I #21 Dec 11 2019 20:09:55

%S 1,2,3,4,5,7,6,8,9,11,15,10,13,14,12,16,17,19,23,31,18,21,27,22,29,30,

%T 20,25,26,28,24,32,33,35,39,47,63,34,37,43,55,38,45,59,46,61,62,36,41,

%U 51,42,53,54,44,57,58,60,40,49,50,52,56,48,64,65,67,71,79,95,127,66,69,75,87,111,70,77,91,119,78,93,123

%N Permutation of natural numbers: a(1) = 1, a(A087686(n)) = 2*a(n-1), a(A088359(n)) = 1+2*a(n), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001, the Hofstadter-Conway $10000 sequence.

%H Antti Karttunen, <a href="/A267111/b267111.txt">Table of n, a(n) for n = 1..8192</a>

%H Antti Karttunen, <a href="/A135141/a135141.pdf">Entanglement Permutations</a>, 2016-2017

%H T. Kubo and R. Vakil, <a href="http://dx.doi.org/10.1016/0012-365X(94)00303-Z">On Conway's recursive sequence</a>, Discr. Math. 152 (1996), 225-252.

%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 2*a(A080677(n)-1), otherwise [when n is in A088359], a(n) = 1 + 2*a(A004001(n)-1).

%F Equally, for n > 1, if A093879(n-1) = 0, a(n) = 2*a(n - A004001(n)), otherwise a(n) = 1 + 2*a(A004001(n)-1). [Above formula in a more symmetric form.]

%F As a composition of other permutations:

%F a(n) = A054429(A276441(n)).

%F a(n) = A233275(A276343(n)).

%F a(n) = A233277(A276345(n)).

%F a(n) = A006068(A276445(n)).

%F Other identities. For all n >= 0:

%F a(2^n) = 2^n. [Follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper.]

%o (Scheme, with memoization-macro definec)

%o (definec (A267111 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (* 2 (A267111 (+ -1 (A080677 n))))) (else (+ 1 (* 2 (A267111 (+ -1 (A004001 n))))))))

%Y Inverse: A267112.

%Y Cf. A004001, A080677, A087686, A088359, A093879, A267108, A267109.

%Y Similar or related permutations: A006068, A054429, A276441, A233275, A233277, A276343, A276345, A276445.

%Y Cf. also permutations A266411, A266412 and arrays A265901, A265903.

%K nonn,look

%O 1,2

%A _Antti Karttunen_, Jan 10 2016