A175498 -id:A175498 - cp's OEIS frontend

A175499 a(n) = A175498(n+1)-A175498(n).

1, 2, -1, 3, 4, -5, 6, -4, 5, -3, 7, -8, 9, -2, 8, -10, 11, -6, 10, -14, 12, -7, 13, -12, 14, -13, 15, -11, 16, -19, 17, -9, 18, -21, 19, -17, 20, -18, 21, -15, 22, -26, 23, -16, 24, -29, 25, -22, 27, -23, 26, -25, 28, -27, 29, -28, 30, -24, 31, -35, 32, 33, -62, 34, -31, 35, -34, 36, -33, 37, -39, 38, -32, 39, -42, 40, -36

Offset: 1

Leroy Quet, May 31 2010

sign

No integer occurs in this sequence more than once, by definition. Is this sequence a permutation of the nonzero integers?

Chai Wah Wu, Table of n, a(n) for n = 1..4999

Cf. A175498, A257884, A131389.

a175499 n = a175499_list !! (n-1)
a175499_list = zipWith (-) (tail a175498_list) a175498_list
-- Reinhard Zumkeller, Apr 25 2015

a[1] = 0; d[1] = 1; 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}]  (* A257884 *)
Table[d[k], {k, 1, zz}] (* A175499 *)
(* Clark Kimberling, May 13 2015 *)

A175499_list, l, s, b = [1], 2, 3, set()
for n in range(2, 10**2):
    i, j = s, s-l
    while True:
        if not (i in b or j in A175499_list):
            A175499_list.append(j)
            b.add(i)
            l = i
            while s in b:
                b.remove(s)
                s += 1
            break
        i += 1
        j += 1 # Chai Wah Wu, Dec 15 2014

More terms from Sean A. Irvine, Jan 27 2011

A257465 Inverse permutation to A175498.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 9, 13, 11, 6, 8, 10, 17, 21, 15, 12, 14, 19, 23, 25, 27, 31, 16, 18, 29, 22, 35, 20, 37, 33, 39, 24, 43, 26, 47, 28, 41, 49, 32, 45, 30, 51, 53, 55, 57, 36, 61, 34, 38, 64, 59, 40, 66, 68, 72, 44, 70, 76, 42, 48, 74, 78, 80, 46, 50, 84, 82

Offset: 1

Reinhard Zumkeller, Apr 25 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A175498.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a257465 = (+ 1) . fromJust . (`elemIndex` a175498_list)

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

Original entry on oeis.org

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

Clark Kimberling, May 13 2015

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

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

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

Cf. A081145, A175498, A257705.

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 *)

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

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

A257705 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 0.

Original entry on oeis.org

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, 26, 14, 29, 45, 31, 16, 33, 51, 35, 54, 37, 57, 38, 59, 41, 63, 43, 22, 46, 24, 47, 72, 49, 75, 50, 77, 53, 81, 55, 28, 58, 87, 56, 88, 60, 91, 62, 95, 65, 99, 67

Offset: 1

Clark Kimberling, May 12 2015

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.
Guide to related sequences:
a(1)  d(1)      (a(n))             (d(n))
0       0      A257705      A131389 except for initial terms
0       1      A257706      A131389 except for initial terms
0       2      A257876      A131389 except for initial terms
0       3      A257877      A257915
1       0      A131388      A131389
1       1      A257878      A131389 except for initial terms
2       0      A257879      A257880
2       1      A257881      A257880 except for initial terms
2       2      A257882      A257918

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

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

Cf. A131388, A081145, A257883, A175498.

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]];
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}] (* A257705 *)
Table[d[k], {k, 1, zz}]     (* A131389 *)

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

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

A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 8, 5, 11, 7, 12, 19, 14, 22, 16, 9, 18, 28, 20, 31, 21, 33, 24, 13, 26, 40, 27, 15, 30, 46, 32, 17, 34, 52, 36, 55, 38, 58, 39, 60, 42, 64, 44, 23, 47, 25, 48, 73, 50, 76, 51, 78, 54, 82, 56, 29, 59, 88, 57, 89, 61, 92, 63, 96, 66, 100, 68, 35, 70, 106, 72, 37

Offset: 1

Clark Kimberling, Jul 05 2007

nonn

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.

			a(2)=1+1, a(3)=a(2)+2, a(4)=a(3)+(-1), a(5)=a(4)+3, a(6)=a(5)+4.

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

Cf. A131389, A131390, A131391, A131392, A131393, A131394, A131395, A131396, A131397, A257705, A175498.

(*Program 1 *)
{a, f} = {{1}, {0}}; Do[tmp = {#, # - Last[a]} &[Max[Complement[#, Intersection[a, #]] &[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f,#] && ! MemberQ[a, Last[a] + #]) &]]];
AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {400}];
{a, f} (*{A131388, A131389}; Peter J. C. Moses, May 10 2015*)
(*Program 2 *)
a[1] = 1; 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]];
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}] (* A131388 *)
Table[d[k], {k, 1, zz}]     (* A131389 *)

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

Revised by Clark Kimberling, May 12 2015

A257905 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 0 and d(1) = 0.

Original entry on oeis.org

0, 1, 3, 2, 5, 11, 4, 9, 6, 13, 7, 15, 10, 8, 17, 35, 12, 25, 14, 29, 16, 33, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 30, 26, 53, 24, 49, 40, 28, 57, 27, 55, 31, 63, 32, 65, 38, 42, 34, 69, 36, 73, 48, 97, 44, 89, 46, 93, 51, 103, 52, 105, 50, 101

Offset: 1

Clark Kimberling, May 16 2015

Rule 3 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 least 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:  suppose that a(1) is an nonnegative integer and d(1) is an integer.
If a(1) = 0 and d(1) != 1, then (a(n)) is a permutation of the nonnegative integers;
if a(1) = 0 and d(1) = 1, then (a(n)) is a permutation of the nonnegative integers excluding 1;
if a(1) = 1, then (a(n)) is a permutation of the positive integers;
if a(1) > 1, then (a(n)) is a permutation of the integers >1;
if d(1) = 0, then (d(n)) is a permutation of the integers;
if d(1) !=0, then (d(n)) is a permutation of the nonzero integers.
Guide to related sequences:
a(1)  d(1)      (a(n))       (d(n))
0       0      A257905      A258047
0       1      A257906      A257907
0       2      A257908      A257909
0       3      A257910      A257980
1       0      A258046      A258047
1       1      A257981      A257982
1       2      A257983      A257909
2       0      A257985      A257047
2       1      A257986      A257982
2       2      A257987      A257909

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

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

Cf. A258047, A257705, A257883, A175498.

Cf. A256283 (putative inverse).

import Data.List ((\\))
a257905 n = a257905_list !! (n-1)
a257905_list = 0 : f [0] [0] where
   f xs@(x:_) ds = g [2 - x .. -1] where
     g [] = y : f (y:xs) (h:ds) where
                  y = x + h
                  (h:_) = [z | z <- [1..] \\ ds, x - z `notElem` xs]
     g (h:hs) | h `notElem` ds && y `notElem` xs = y : f (y:xs) (h:ds)
              | otherwise = g hs
              where y = x + h
-- Reinhard Zumkeller, Jun 03 2015

{a, f} = {{0}, {0}}; Do[tmp = {#, # - Last[a]} &[Min[Complement[#, Intersection[a, #]]&[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f, #] && ! MemberQ[a, Last[a] - #]) &]]]; AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {120}]; {a, f} (* Peter J. C. Moses, May 14 2015 *)

a(n) = A258046(n) - 1 for n >= 1.

A327743 a(n) = smallest positive number not already in the sequence such that for each k = 1, ..., n-1, the k-th differences are distinct.

Original entry on oeis.org

1, 2, 4, 3, 6, 11, 5, 9, 7, 13, 10, 18, 8, 15, 27, 14, 23, 12, 22, 17, 28, 16, 29, 20, 34, 19, 35, 21, 36, 32, 24, 42, 26, 43, 25, 44, 66, 33, 53, 30, 51, 31, 54, 37, 61, 39, 64, 38, 67, 40, 70, 41, 68, 47, 75, 50, 76, 45, 77, 49, 80, 48, 81, 46, 82, 52, 86

Offset: 1

Peter Kagey, Sep 24 2019

Is this sequence a permutation of the positive integers?
Does each k-th difference contain all nonzero integers?
It is not difficult to show that if a(1), ..., a(k) satisfy the requirements, then any sufficiently large number is a candidate for a(k+1). So a(k) exists for all k. - N. J. A. Sloane, Sep 24 2019
The original definition was "Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, the k-th differences are distinct."
If only first differences are considered, one gets the classical Mian-Chowla sequence A005282. - M. F. Hasler, Oct 09 2019

			Illustration of the first eight terms of the sequence.
k | k-th differences
--+---------------------------------
0 |   1,  2,   4,   3,   6, 11, 5, 9
1 |   1,  2,  -1,   3,   5, -6, 4
2 |   1, -3,   4,   2, -11, 10
3 |  -4,  7,  -2, -13,  21
4 |  11, -9, -11,  34
5 | -20, -2,  45
6 |  18, 47
7 |  29

Peter Kagey, Table of n, a(n) for n = 1..5000

Cf. A175498.
First differences: A327452; leading column of difference triangle: A327457.
If ALL terms of the difference triangle must be distinct, see A327460 and A327762.
Cf. A005282.

a[1] = 1;
a[n_] := a[n] = For[aa = Array[a, n-1]; an = 1, True, an++, If[FreeQ[aa, an], aa = Append[aa, an]; If[AllTrue[Range[n-1], Unequal @@ Differences[ aa, #]&], Return[an]]]];
a /@ Range[1, 100] (* Jean-François Alcover, Oct 26 2019 *)

"Infinite" added to definition (for otherwise the one-term sequence 1 is earlier). - N. J. A. Sloane, Sep 25 2019

Changed definition to avoid use of "Lexicographically earliest infinite sequence" and the associated existence questions. - N. J. A. Sloane, Sep 28 2019

A231334 Lexicographically earliest sequence of distinct positive integers such that for any distinct i,j, k, the points at positions (i, a(i)), (j, a(j)), (k, a(k)) are not aligned.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 12, 7, 14, 13, 8, 23, 17, 18, 22, 10, 15, 11, 28, 19, 16, 20, 29, 32, 44, 35, 39, 24, 40, 26, 37, 42, 21, 56, 64, 43, 31, 25, 34, 27, 33, 66, 67, 52, 60, 30, 57, 36, 63, 86, 82, 38, 50, 47, 69, 75, 79, 89, 49, 45, 76, 41, 48, 98, 77, 94

Offset: 1

Paul Tek, Nov 07 2013

nonn

Is this a permutation of the natural numbers?

There are only two fixed points: 1 and 2.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, C program for this sequence
Illustrations of the first 200 points at positions (n, a(n)) (black pixels correspond to these points, colored pixels (x,y) are aligned with two black pixels (i,a(i)) and (j,a(j))):
(1) x < i < j,
(2) i < x < j,
(3) i < j < x.

Cf. A175498, A236266, A236335, A300002.

C
```
See Link section.
```

WIDTH = 1000;
HEIGHT = 2000;
Clear[seen, aligned, a];
compute[n_] := Module[{c = 1}, While[seen[c] || aligned[n][c], c++; If[c > HEIGHT, Abort[]]]; a[n] = c; seen[a[n]] = True; For[i = 1, i < n, i++, dn = n - i; da = a[n] - a[i]; g = GCD[dn, da]; dn /= g; da /= g; nn = n; na = c; While[True, nn += dn; If[nn > WIDTH, Break[]]; na += da; If[na < 1 || na > HEIGHT, Break[]]; aligned[nn][na] = True]]; a[n]];
Array[compute, WIDTH] (* Jean-François Alcover, Apr 19 2020, translated from Paul Tek's program. *)

A257218 Lexicographically earliest sequence of distinct positive integers such that gcd(a(n), a(n-1)) takes no value more than twice.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 10, 5, 15, 9, 18, 12, 16, 24, 30, 20, 40, 32, 48, 36, 27, 54, 72, 60, 45, 75, 25, 50, 70, 7, 14, 28, 42, 21, 63, 126, 84, 56, 112, 64, 96, 120, 80, 100, 150, 90, 108, 81, 162, 216, 144, 168, 140, 35, 105, 210, 180, 135, 225, 300

Offset: 1

Ivan Neretin, Apr 18 2015

Presumably a(n) is a permutation of the positive integers.
Primes seem to occur in their natural order. 31 appears as a(7060). Primes p >= 37 are not found among the first 10000 terms.
Numbers n such that a(n)=n are 1, 2, 3, 12, 306, ...
A256918(n) = gcd(a(n), a(n+1)); gcd(a(A257120(n)), a(A257120(n)+1)) = gcd(a(A257475(n)), a(A257475(n)-1)) = n. - Reinhard Zumkeller, Apr 25 2015
For p prime: A257122(p)-1 = index of the smallest multiple of p: a(A257122(p)-1) mod p = 0 and a(m) mod p > 0 for m < A257122(p)-1. - Reinhard Zumkeller, Apr 26 2015

			After a(9)=15, the values 1, 2, 3, 4, 6, and 8 are already used, while 7 is forbidden because gcd(15,7)=1 and that value of GCD has already occurred twice, at (1,2) and (2,3). The minimal value which is neither used not forbidden is 9, so a(10)=9.

Ivan Neretin and Reinhard Zumkeller, Table of n, a(n) for n = 1..25000, first 10000 terms from Ivan Neretin

Other minimal sequences of distinct positive integers that match some condition imposed on a(n) and a(n-1):
A175498 (differences are unique),
A081145 (absolute differences are unique),
A235262 (bitwise XORs are unique),
A163252 (differ by one bit in binary),
A000027 (GCD=1),
A064413 (GCD>1),
A128280 (sum is a prime),
A034175 (sum is a square),
A175428 (sum is a cube),
A077220 (sum is a triangular number),
A073666 (product plus 1 is a prime),
A081943 (product minus 1 is a prime),
A091569 (product plus 1 is a square),
A100208 (sum of squares is a prime).
Cf. A004526.
Cf. A256918, A257120, A257475, A257478, A257122 (putative inverse).
Cf. also A281978.

import Data.List (delete); import Data.List.Ordered (member)
a257218 n = a257218_list !! (n-1)
a257218_list = 1 : f 1 [2..] a004526_list where
   f x zs cds = g zs where
     g (y:ys) | cd `member` cds = y : f y (delete y zs) (delete cd cds)
              | otherwise       = g ys
              where cd = gcd x y
-- Reinhard Zumkeller, Apr 24 2015

a={1}; used=Array[0&,10000]; Do[i=1; While[MemberQ[a,i] || used[[l=GCD[a[[-1]],i]]]>=2, i++]; used[[l]]++; AppendTo[a,i], {n,2,100}]; a (* Ivan Neretin, Apr 18 2015 *)

A257706 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 1.

Original entry on oeis.org

0, 2, 1, 4, 8, 6, 3, 9, 5, 10, 17, 12, 20, 14, 7, 16, 26, 18, 29, 19, 31, 22, 11, 24, 38, 25, 13, 28, 44, 30, 15, 32, 50, 34, 53, 36, 56, 37, 58, 40, 62, 42, 21, 45, 23, 46, 71, 48, 74, 49, 76, 52, 80, 54, 27, 57, 86, 55, 87, 59, 90, 61, 94, 64, 98, 66, 33

Offset: 1

Clark Kimberling, May 12 2015

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.

			a(1) = 0, d(1) = 1;
a(2) = 2, d(2) = 2;
a(3) = 1, d(3) = -1;
a(4) = 4, d(4) = 3;
(The sequence d differs from A131389 only in the first 13 terms.)

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

Cf. A131388, A257705, A081145, A257883, A175498.

a[1] = 0; d[1] = 1; 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}]  (* A257706 *)
Table[d[k], {k, 1, zz}]  (* A131389 shifted *)

a(n+1) - a(n) = d(n+1) = A131389(n+1) for n >= 1.

cp's OEIS Frontend

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