A257902 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.

2, 1, 3, -2, 4, -3, 5, -1, 6, -8, 7, 8, -11, 9, -7, 10, -6, -5, 12, 11, -19, 13, -10, 14, -15, 16, -13, 15, -9, 17, -18, 19, -20, 18, -12, 20, -23, 21, -17, 22, -24, 23, -16, 24, -22, 25, -26, 27, -25, 26, -14, -4, 28, -35, 29, -28, 30, 31, -57, 32, -31, 33

Offset: 1

Author

Clark Kimberling, May 13 2015

Keywords

• easy
• sign

Comments

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.

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 A257883 for a guide to related sequences.

Examples

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

Links

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

Crossrefs

Cf. A257885, A257883, A257705.

Programs

Mathematica

a[1] = 0; d[1] = 2; 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]]
Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
   d[k + 1] = h, k = k + 1}, {i, 1, zz}];
u = Table[a[k], {k, 1, zz}]  (* A257885 *)
Table[d[k], {k, 1, zz}]      (* A257902 *)

Formula

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