A257911 - OEIS

%I #6 Jun 16 2015 13:53:41

%S 2,1,4,5,3,7,12,6,13,8,14,10,18,9,19,11,20,17,28,15,27,16,29,22,36,21,

%T 37,23,38,26,43,24,42,25,44,34,54,30,51,31,53,32,55,33,57,39,64,35,61,

%U 45,72,40,68,41,70,47,77,46,78,48,79,112,49,83,50,85,59

%N Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 2 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="/A257911/b257911.txt">Table of n, a(n) for n = 1..1000</a>

%t a[1] = 2; 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, d[k + 1] = h, k = k + 1}, {i, 1, zz}];

%t Table[a[k], {k, 1, zz}]  (* A257911 *)

%t Table[d[k], {k, 1, zz}]  (* A257912 *)

%Y Cf. A257912, A257883.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 12 2015