cp's OEIS Frontend

A257904 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.

%I A257904 #9 Jun 11 2015 10:35:49
%S A257904 3,1,2,-1,4,-2,5,-4,6,-3,7,-8,9,-6,8,-5,10,-11,12,-10,11,13,-21,14,
%T A257904 -12,15,-14,16,-15,18,-17,19,-13,17,-19,20,-16,21,-24,22,-20,23,-9,24,
%U A257904 -35,25,-22,26,-28,27,-23,28,-25,29,-27,30,-18,-7,31,-34,32,-26
%N A257904 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.
%C A257904 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 A257904 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 A257904 See A257883 for a guide to related sequences.
%H A257904 Clark Kimberling, <a href="/A257904/b257904.txt">Table of n, a(n) for n = 1..1000</a>
%F A257904 a(k+1) - a(k) = d(k+1) for k >= 1.
%e A257904 a(1) = 0, d(1) = 2;
%e A257904 a(2) = 1, d(2) = 1;
%e A257904 a(3) = 4, d(3) = 3;
%e A257904 a(4) = 2, d(4) = -2.
%t A257904 a[1] = 0; d[1] = 3; k = 1; z = 10000; zz = 120;
%t A257904 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257904 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257904 T[k_] := -a[k] + Complement[Range[z], A[k]]
%t A257904 Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
%t A257904    d[k + 1] = h, k = k + 1}, {i, 1, zz}];
%t A257904 u = Table[a[k], {k, 1, zz}]  (* A257903 *)
%t A257904 Table[d[k], {k, 1, zz}]      (* A257904 *)
%Y A257904 Cf. A257903, A257883, A257705.
%K A257904 easy,sign
%O A257904 1,1
%A A257904 _Clark Kimberling_, May 13 2015