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A138606 List first F(1) odd numbers, then first F(2) even numbers (starting from 2), then the next F(3) odd numbers, then the next F(4) even numbers, etc., where F(n) = A000045(n), the n-th Fibonacci number.

%I A138606 #20 Sep 05 2017 04:04:29
%S A138606 1,2,3,5,4,6,8,7,9,11,13,15,10,12,14,16,18,20,22,24,17,19,21,23,25,27,
%T A138606 29,31,33,35,37,39,41,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,
%U A138606 58,60,62,64,66,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77
%N A138606 List first F(1) odd numbers, then first F(2) even numbers (starting from 2), then the next F(3) odd numbers, then the next F(4) even numbers, etc., where F(n) = A000045(n), the n-th Fibonacci number.
%C A138606 The original name was "FibCon sequence". However, this sequence has only a passing resemblance to Connell-like sequences (see A001614), which are all monotone, while this sequence is a bijection of natural numbers.
%C A138606 Fixed points of the permutation are the terms of A062114. - _Ivan Neretin_, Sep 04 2017
%H A138606 Ivan Neretin, <a href="/A138606/b138606.txt">Table of n, a(n) for n = 1..10000</a>
%H A138606 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A138606 a(n) = A166012(A072649(n)-1) + 2*(n - A000045(1+A072649(n))). - _Antti Karttunen_, Oct 05 2009
%e A138606 Let us separate the positive integers into odd (A005408) and even numbers (A005843):
%e A138606 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,...
%e A138606 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,...
%e A138606 then we get the following subsequences:
%e A138606 S1={1}
%e A138606 S2={2}
%e A138606 S3={3,5}
%e A138606 S4={4,6,8}
%e A138606 S5={7,9,11,13,15}
%e A138606 S6={10,12,14,16,18,20,22,24}
%e A138606 ...
%e A138606 and concatenating them S1/S2/S3/S4/S5/... gives this sequence.
%t A138606 o = 1; e = 2; Flatten@Table[If[OddQ[n], Range[o, (o += 2 Fibonacci[n]) - 1, 2], Range[e, (e += 2 Fibonacci[n]) - 1, 2]], {n, 9}] (* _Ivan Neretin_, Sep 04 2017 *)
%o A138606 (MIT Scheme:) (define (A138606 n) (if (zero? n) n (+ (A166012 (-1+ (A072649 n))) (* 2 (- n (A000045 (1+ (A072649 n))))))))
%Y A138606 Inverse: A166013. A000035(a(n)) = A000035(A072649(n)). Cf. A138607-A138609, A138612.
%K A138606 easy,nonn
%O A138606 1,2
%A A138606 _Ctibor O. Zizka_, May 14 2008
%E A138606 Edited, extended and Scheme code added by _Antti Karttunen_, Oct 05 2009