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%I #23 Jun 19 2021 02:36:27
%S 1,2,5,3,7,4,10,6,14,8,17,9,20,11,23,12,25,13,28,15,31,16,34,18,37,19,
%T 40,21,43,22,46,24,50,26,53,27,56,29,59,30,62,32,65,33,68,35,71,36,74,
%U 38,77,39,80,41,83,42,86,44,89,45,92,47,95,48,97,49,100
%N a(1) = 1, a(2) = 2; for n > 1, a(2*n+2) = smallest number not yet seen, a(2*n+1) = a(2*n) + a(2*n+2).
%C Permutation of natural numbers with inverse A210771.
%C From _Jeffrey Shallit_, Jun 18 2021: (Start)
%C This sequence is "2-sychronized"; there is a 23-state finite automaton that recognizes the base-2 representations of n and a(n), in parallel.
%C It obeys the identities
%C a(4n+3) = a(2n+1) - a(4n) + 2 a(4n+2)
%C a(8n) = 2a(4n)
%C a(8n+1) = a(2n+1) + 3a(4n)
%C a(8n+2) = a(2n+1) + 2 a(4n) - a(4n+1) + a(4n+2)
%C a(8n+4) = a(2n+1) + a(4n+2)
%C a(8n+5) = 3a(2n+1) - a(4n) +2a(4n+2)
%C a(8n+6) = 2a(2n+1) - a(4n) + a(4n+2). (End)
%H Reinhard Zumkeller, <a href="/A210770/b210770.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Per#IntegerPermutation"> Index entries for sequences that are permutations of the natural numbers</a>
%F a(2*n-1) = A022441(n-1); a(2*n) = A055562(n-1).
%o (Haskell)
%o import Data.List (delete)
%o a210770 n = a210770_list !! (n-1)
%o a210770_list = 1 : 2 : f 1 2 [3..] where
%o f u v (w:ws) = u' : w : f u' w (delete u' ws) where u' = v + w
%o (Python)
%o def aupton(terms):
%o alst, seen = [1, 2], {1, 2}
%o for n in range(2, terms, 2):
%o anp1 = alst[-1] + 1
%o while anp1 in seen: anp1 += 1
%o an = alst[n-1] + anp1
%o alst, seen = alst + [an, anp1], seen | {an, anp1}
%o return alst[:terms]
%o print(aupton(67)) # _Michael S. Branicky_, Jun 18 2021
%Y Cf. A064736.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Mar 25 2012
%E Definition corrected by _Jeffrey Shallit_, Jun 18 2021
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