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%I #10 Jun 09 2015 14:21:59
%S 2,1,3,-2,4,-3,5,-1,6,-8,7,8,-11,9,-7,10,-6,-5,12,11,-19,13,-10,14,
%T -15,16,-13,15,-9,17,-18,19,-20,18,-12,20,-23,21,-17,22,-24,23,-16,24,
%U -22,25,-26,27,-25,26,-14,-4,28,-35,29,-28,30,31,-57,32,-31,33
%N Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.
%C Algorithm: 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). Let h be the least integer > -a(k) such that h is not in D(k) and 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 repeat inductively.
%C Conjecture: if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0). Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.
%C See A257883 for a guide to related sequences.
%H Clark Kimberling, <a href="/A257902/b257902.txt">Table of n, a(n) for n = 1..1000</a>
%F a(k+1) - a(k) = d(k+1) for k >= 1.
%e a(1) = 0, d(1) = 2;
%e a(2) = 1, d(2) = 1;
%e a(3) = 4, d(3) = 3;
%e a(4) = 2, d(4) = -2.
%t a[1] = 0; d[1] = 2; k = 1; z = 10000; zz = 120;
%t A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t c[k_] := Complement[Range[-z, z], diff[k]];
%t T[k_] := -a[k] + Complement[Range[z], A[k]]
%t Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
%t d[k + 1] = h, k = k + 1}, {i, 1, zz}];
%t u = Table[a[k], {k, 1, zz}] (* A257885 *)
%t Table[d[k], {k, 1, zz}] (* A257902 *)
%Y Cf. A257885, A257883, A257705.
%K easy,sign
%O 1,1
%A _Clark Kimberling_, May 13 2015
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