A257918 - OEIS

%I #6 May 14 2015 12:00:42

%S 2,-1,3,1,-2,4,5,-3,6,-4,-5,7,8,-7,-6,9,10,-8,-9,12,11,-10,13,-11,14,

%T -13,15,-12,16,-14,-15,18,-16,17,19,-17,20,-19,-18,22,21,-20,23,-22,

%U 24,-21,-23,25,26,-24,27,-25,28,-26,-27,30,-28,29,31,-29,32,-31

%N Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2.

%C This is the sequence (d(n)) of differences associated with the sequence a = A257882.

%C Rule 1 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 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 greatest such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.

%C 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 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 A257705 for a guide to related sequences.

%H Clark Kimberling, <a href="/A257918/b257918.txt">Table of n, a(n) for n = 1..1000</a>

%F d(k) = a(k) - a(k-1) for k >=2, where a(k) = A257882(k).

%e a(1) = 2, d(1) = 2;

%e a(2) = 1, d(2) = -1;

%e a(3) = 4, d(3) = 3;

%e a(4) = 5, d(4) = 1.

%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 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];

%t Table[If[Length[s[k]] == 0, {h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {h = Max[s[k]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}], {i, 1, zz}];

%t u = Table[a[k], {k, 1, zz}]  (* A257882 *)

%t Table[d[k], {k, 1, zz}]      (* A257918 *)

%Y Cf. A131389, A257705, A081145, A257918, A175499.

%K easy,sign

%O 1,1

%A _Clark Kimberling_, May 13 2015