A257883 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 0.

0, 1, 3, 2, 5, 9, 4, 10, 6, 11, 8, 15, 7, 16, 14, 22, 12, 23, 17, 27, 13, 25, 18, 31, 19, 33, 20, 35, 24, 40, 21, 38, 29, 47, 26, 45, 28, 48, 30, 51, 36, 58, 32, 55, 39, 63, 34, 59, 37, 64, 41, 67, 42, 70, 43, 72, 44, 74, 50, 81, 46, 78, 111, 49, 83, 52, 87

Offset

1,3

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 a 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.

Guide to related sequences:

a(1) d(1) (a(n)) (d(n))

0 0 A257883 A175499 except for initial terms

0 1 A257884 A175499

0 2 A257885 A257902

0 3 A257903 A257904

1 0 A175498 A175499 except for first term

1 1 A257905 A175499

2 0 A257908 A257909

2 1 A257910 A257909 except for initial terms

2 2 A257911 A257912

Links

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

Formula

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

Also, A257883(n) = -1 + A175498(n) for n >= 1.

Example

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

Mathematica

a[1] = 0; d[1] = 0; 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}]  (* A257883, = -1 + A175498 *)
Table[d[k], {k, 1, zz}] (* A175499 except that here first term is 0 *)

Crossrefs

Cf. A081145, A175498, A257705.

Sequence in context: A243786 A278743 A081974 * A169755 A169748 A280875

Adjacent sequences: A257880 A257881 A257882 * A257884 A257885 A257886

Keyword

nonn,easy

Author

Clark Kimberling, May 13 2015

Status

approved