A257915 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 3.

3, 1, 2, -1, 4, -2, 5, -4, 6, -3, 7, -5, 8, -6, 9, -7, 10, -8, -9, 12, 11, -10, 13, -11, 14, -13, 15, -12, 16, -14, -15, 18, 17, -16, 19, -17, 20, -19, -18, 22, 21, -20, 23, -22, 24, -21, 25, -23, 26, -25, 27, -24, 28, -26, -27, 30, -28, 29, 31, -29, 32, -31

Offset: 1

Author

Clark Kimberling, May 12 2015

Keywords

• easy
• sign

Comments

This is the sequence (d(n)) of differences associated with the sequence a = A257877.

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).

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.

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.

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.

See A257705 for a guide to related sequences.

Examples

			a(1) = 0, d(1) = 3;
a(2) = 1, d(2) = 1;
a(3) = 3, d(3) = 2;
a(4) = 2, d(4) = -1.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A131389, A257705, A081145, A257883, A175499.

Programs

Mathematica

a[1] = 0; d[1] = 3; k = 1; z = 10000; zz = 120;
A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
c[k_] := Complement[Range[-z, z], diff[k]];
T[k_] := -a[k] + Complement[Range[z], A[k]];
s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
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}];
u = Table[a[k], {k, 1, zz}]  (* A257877 *)
Table[d[k], {k, 1, zz}]  (* A257915 *)

Formula

d(k) = a(k) - a(k-1) for k >= 2, where a(k) = A257877(k).