

A257902


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


3



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
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OFFSET

1,1


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.


LINKS

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


FORMULA

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


EXAMPLE

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


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 *)


CROSSREFS

Cf. A257885, A257883, A257705.
Sequence in context: A283994 A026238 A066136 * A257909 A289439 A213633
Adjacent sequences: A257899 A257900 A257901 * A257903 A257904 A257905


KEYWORD

easy,sign


AUTHOR

Clark Kimberling, May 13 2015


STATUS

approved



