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%I #37 May 30 2016 18:22:58
%S 0,1,4,2,18,6,12,3,96,24,48,8,72,15,16,5,600,120,240,30,360,56,60,10,
%T 480,87,88,20,90,21,22,7,4320,720,1440,144,2160,270,288,36,2880,416,
%U 420,67,432,73,66,13,3600,567,568,107,570,109,108,26,576,111,112,27,114,28,52,9,35280,5040,10080,840,15120,1584,1680,168
%N a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).
%C This sequence can be represented as a binary tree. Each left hand child is produced as A255411(n), and each right hand child as A256450(n), when parent contains n >= 1:
%C 0
%C |
%C ...................1...................
%C 4 2
%C 18......../ \........6 12......../ \........3
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C 96 24 48 8 72 15 16 5
%C 600 120 240 30 360 56 60 10 480 87 88 20 90 21 22 7
%C etc.
%C Because all terms of A255411 are even it means that odd terms can occur only in odd positions (together with some even terms, for each one of which there is a separate infinite cycle), while terms in even positions are all even.
%C After its initial 1, A255567 seems to give all the terms like 2, 3, 12, ... where the left hand child of the right hand child is one more than the right hand child of the left hand child (as for 2: 16 = 15+1, as for 3: 22 = 21+1, as for 12: 88 = 87+1).
%H Antti Karttunen, <a href="/A255566/b255566.txt">Table of n, a(n) for n = 0..1023</a>
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).
%F Other identities:
%F For all n >= 0, a(2^n) = A001563(n+1). [The leftmost branch of the binary tree is given by n*n!]
%F For all n >= 0, a(A083318(n)) = A000142(n+1). [And the next innermost vertices by (n+1)! This follows because A256450(n*n! - 1) = (n+1)! - 1.]
%F For all n >= 1, A257679(a(n)) = A001511(n).
%o (Scheme, with memoization-macro definec)
%o (definec (A255566 n) (cond ((zero? n) n) ((even? n) (A255411 (A255566 (/ n 2)))) (else (A256450 (A255566 (/ (- n 1) 2))))))
%Y Inverse: A255565.
%Y Cf. A000142, A001511, A001563, A083318, A255411, A256450, A257679.
%Y Cf. also A255567 and arrays A257503, A257505.
%Y Related or similar permutations: A273666, A273667.
%K nonn,base,tabf
%O 0,3
%A _Antti Karttunen_, May 05 2015
%E Formula changed because of the changed starting offset of A256450 - _Antti Karttunen_, May 30 2016
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