A257705 - OEIS

%I #9 May 14 2015 11:58:54

%S 0,1,3,2,5,9,7,4,10,6,11,18,13,21,15,8,17,27,19,30,20,32,23,12,25,39,

%T 26,14,29,45,31,16,33,51,35,54,37,57,38,59,41,63,43,22,46,24,47,72,49,

%U 75,50,77,53,81,55,28,58,87,56,88,60,91,62,95,65,99,67

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

%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 Guide to related sequences:

%C a(1)  d(1)      (a(n))             (d(n))

%C 0       0      A257705      A131389 except for initial terms

%C 0       1      A257706      A131389 except for initial terms

%C 0       2      A257876      A131389 except for initial terms

%C 0       3      A257877      A257915

%C 1       0      A131388      A131389

%C 1       1      A257878      A131389 except for initial terms

%C 2       0      A257879      A257880

%C 2       1      A257881      A257880 except for initial terms

%C 2       2      A257882      A257918

%H Clark Kimberling, <a href="/A257705/b257705.txt">Table of n, a(n) for n = 1..1000</a>

%F a(k+1) - a(k) = d(k+1) for k >= 1.

%F Also, a(k) = A131388(n)-1.

%e a(2) = a(1) + d(2) = 0 + 1 = 1;

%e a(3) = a(2) + d(3) = 1 + 2 = 3;

%e a(4) = a(3) + d(4) = 3 + (-1) = 2.

%t a[1] = 0; d[1] = 0; 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}] (* A257705 *)

%t Table[d[k], {k, 1, zz}]     (* A131389 *)

%Y Cf. A131388, A081145, A257883, A175498.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, May 12 2015