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A257906 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 0 and d(1) = 1.

%I A257906 #18 Jun 03 2015 10:58:05
%S A257906 0,2,5,3,7,4,9,8,15,6,14,10,19,11,21,16,31,12,23,13,25,18,35,17,33,20,
%T A257906 39,22,43,27,53,24,47,26,51,28,55,29,57,30,59,34,67,32,63,41,81,36,71,
%U A257906 37,73,40,79,38,75,44,87,45,89,42,83,46,91,48,95,49,97
%N A257906 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 0 and d(1) = 1.
%C A257906 Rule 3 follows.  For k >= 1, let  A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).
%C A257906 Step 1:  If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the least such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
%C A257906 Step 2:  Let h be the least positive integer not in D(k) such that a(k) - h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.
%C A257906 See A257905 for a guide to related sequences and conjectures.
%H A257906 Clark Kimberling (first 1000 terms) & Antti Karttunen, <a href="/A257906/b257906.txt">Table of n, a(n) for n = 1..10000</a>
%e A257906 a(1) = 0, d(1) = 1;
%e A257906 a(2) = 2, d(2) = 2;
%e A257906 a(3) = 5, d(3) = 3;
%e A257906 a(4) = 3, d(4) = -2.
%t A257906 {a, f} = {{0}, {1}}; Do[tmp = {#, # - Last[a]} &[Min[Complement[#, Intersection[a, #]]&[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
%t A257906 If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f, #] && ! MemberQ[a, Last[a] - #]) &]]]; AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {120}]; {a, f} (* _Peter J. C. Moses_, May 14 2015 *)
%o A257906 (Scheme, with memoization-macro definec)
%o A257906 (definec (A257906 n) (if (= 1 n) 0 (+ (A257906 (- n 1)) (A257907 n))))
%o A257906 ;; Needs also code from A257907. - _Antti Karttunen_, May 20 2015
%o A257906 (Haskell)
%o A257906 import Data.List ((\\))
%o A257906 a257906 n = a257906_list !! (n-1)
%o A257906 a257906_list = 0 : f [0] [1] where
%o A257906    f xs@(x:_) ds = g [2 - x .. -1] where
%o A257906      g [] = y : f (y:xs) (h:ds) where
%o A257906                   y = x + h
%o A257906                   (h:_) = [z | z <- [1..] \\ ds, x - z `notElem` xs]
%o A257906      g (h:hs) | h `notElem` ds && y `notElem` xs = y : f (y:xs) (h:ds)
%o A257906               | otherwise = g hs
%o A257906               where y = x + h
%o A257906 -- _Reinhard Zumkeller_, Jun 03 2015
%Y A257906 Cf. A257907 (the associated d-sequence, 1 followed by the first differences of this sequence).
%Y A257906 Cf. A258105 (inverse to a conjectured permutation of natural numbers which is obtained from this sequence by substituting 1 for the initial value a(1) instead of 0).
%Y A257906 Cf. also A257905.
%K A257906 nonn,easy
%O A257906 1,2
%A A257906 _Clark Kimberling_, May 16 2015