A257883 -id:A257883 - cp's OEIS frontend

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

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 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.

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

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

Cf. A257885, A257883, A257705.

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 *)

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

A257903 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 3.

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 9, 5, 11, 8, 15, 7, 16, 10, 18, 13, 23, 12, 24, 14, 25, 38, 17, 31, 19, 34, 20, 36, 21, 39, 22, 41, 28, 45, 26, 46, 30, 51, 27, 49, 29, 52, 43, 67, 32, 57, 35, 61, 33, 60, 37, 65, 40, 69, 42, 72, 54, 47, 78, 44, 76, 50, 83, 53, 87, 48, 84

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 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.

			a(1) = 0, d(1) = 3;
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. A257904, A257883, A257705.

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]]
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}]  (* A257903 *)
Table[d[k], {k, 1, zz}]      (* A257904 *)

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

A257904 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.

Original entry on oeis.org

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

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 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.

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

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

Cf. A257903, A257883, A257705.

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]]
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}]  (* A257903 *)
Table[d[k], {k, 1, zz}]      (* A257904 *)

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

A257911 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 2 and d(1) = 2.

Original entry on oeis.org

2, 1, 4, 5, 3, 7, 12, 6, 13, 8, 14, 10, 18, 9, 19, 11, 20, 17, 28, 15, 27, 16, 29, 22, 36, 21, 37, 23, 38, 26, 43, 24, 42, 25, 44, 34, 54, 30, 51, 31, 53, 32, 55, 33, 57, 39, 64, 35, 61, 45, 72, 40, 68, 41, 70, 47, 77, 46, 78, 48, 79, 112, 49, 83, 50, 85, 59

Offset: 1

Clark Kimberling, Jun 12 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 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.

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

Cf. A257912, A257883.

a[1] = 2; 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}];
Table[a[k], {k, 1, zz}]  (* A257911 *)
Table[d[k], {k, 1, zz}]  (* A257912 *)

A257912 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 2 and d(1) = 2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 12 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 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.

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

Cf. A257911, A257883.

a[1] = 2; 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}];
Table[a[k], {k, 1, zz}]  (* A257911 *)
Table[d[k], {k, 1, zz}]  (* A257912 *)

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

Original entry on oeis.org

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

Clark Kimberling, May 12 2015

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.

			a(1) = 0, d(1) = 3;
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. A131389, A257705, A081145, A257883, A175499.

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 *)

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

A280875 Set a(1)=0, a(2)=1, a(3)=3; b(1)=1, b(2)=2; c(1)=3. Thereafter, a(n) is the smallest positive integer m such that m is not yet in sequence a, m-a(n-1) is not yet in sequence b, and m-a(n-2) is not yet in sequence c; set b(n-1)=m-a(n-1), c(n-2)=m-a(n-2).

Original entry on oeis.org

0, 1, 3, 2, 5, 9, 4, 13, 10, 6, 18, 11, 16, 7, 25, 17, 15, 28, 12, 19, 27, 8, 14, 24, 35, 49, 20, 37, 52, 21, 44, 33, 23, 47, 41, 29, 54, 70, 22, 42, 61, 36, 78, 31, 53, 74, 30, 57, 83, 26, 56, 84, 32, 64, 51, 34, 71, 100, 38, 81, 60, 40, 86, 63, 39, 87, 69, 43

Offset: 1

Luca Petrone, May 14 2018

nonn

			For n=4: m=2 works, because 2 is not in a, 2-3=-1 is not in b, and 2-1=1 is not in c; set a(4)=2, b(3)=-1 and c(2)=1.
For n=5: m=5 works, because 5 is not in a, 5-2=3 is not in b, and 5-3=2 is not in c; set a(5)=5, b(4)=3 and c(3)=2.

N. J. A. Sloane, Table of n, a(n) for n = 1..999
N. J. A. Sloane, Table of [n, a(n), b(n), c(n)] for n = 1..997
Index entries for sequences that are permutations of the natural numbers

Cf. A257883, A308000 (b), A308001 (c).

a = {0, 1};
d1 = {1};
d2 = {};
For[n = 3, n <= 10000, n++,
For[t = Min[Complement[Range[Max[n]], a]], t <= Infinity, t++,
If[MemberQ[a, t] == False,
If[MemberQ[d1, t - a[[n - 1]]] == False && MemberQ[d2, t - a[[n - 2]]] == False,Break[];]]];
a = Flatten[Append[a, t]];
d1 = Flatten[Append[d1, t - a[[n - 1]]]];
d2 = Flatten[Append[d2, t - a[[n - 2]]]];]

Edited by N. J. A. Sloane, Jun 25 2018.

A257876 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 2.

Original entry on oeis.org

0, 1, 4, 3, 7, 5, 2, 8, 13, 9, 16, 11, 19, 12, 6, 15, 25, 17, 28, 18, 30, 21, 10, 23, 37, 24, 39, 27, 43, 29, 14, 31, 49, 33, 52, 35, 55, 36, 57, 34, 56, 38, 61, 41, 20, 44, 22, 47, 73, 48, 75, 51, 79, 53, 26, 58, 87, 59, 89, 60, 91, 54, 88, 50, 83, 42, 77

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) = 2;
a(2) = 1, d(2) = 1;
a(3) = 3, d(3) = 3;
a(4) = 4, d(4) = -1.
The first terms of (d(n)) are (2,1,3,-1,4,-2,-3,6,5,...), which differs from A131389 only in initial terms.

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

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

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]];
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}]  (* A257876 *)
Table[d[k], {k, 1, zz}]      (* A131389 essentially *)

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

A257878 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 1.

Original entry on oeis.org

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, 34

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.
Considering the first 1000 elements of this sequence and A257705 it appears that this is the same as A257705 apart from an index shift. - R. J. Mathar, May 14 2015

			a(1) = 1, d(1) = 1;
a(2) = 3, d(2) = 2;
a(3) = 2, d(3) = -1;
a(4) = 5, d(4) = -3.

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

Cf. A131388, A131389, A257705, A081145, A257883, A175498.

a[1] = 1; 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}]  (* A257878 *)
Table[d[k], {k, 1, zz}]      (* A131389 essentially *)

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

A257881 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 1.

Original entry on oeis.org

2, 1, 3, 6, 4, 8, 5, 10, 16, 12, 7, 14, 22, 15, 9, 18, 28, 20, 11, 23, 13, 24, 37, 26, 40, 27, 42, 30, 46, 32, 17, 34, 52, 36, 19, 38, 58, 39, 21, 43, 64, 44, 67, 45, 69, 48, 25, 50, 76, 51, 78, 54, 82, 56, 29, 59, 31, 60, 91, 62, 94, 63, 33, 66, 100, 68, 35

Offset: 1

Clark Kimberling, May 13 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) = 2, d(1) = 0;
a(2) = 1, d(2) = -1;
a(3) = 3, d(3) = 2;
a(4) = 6, d(4) = 3.

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

Cf. A257880, A257705, A081145, A257883, A175498.

a[1] = 2; 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}]  (* A257881 *)
Table[d[k], {k, 1, zz}]  (* essentially A257880 *)

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

cp's OEIS Frontend

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

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A257903 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 3.

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A257904 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.

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A257911 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 2 and d(1) = 2.

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A257912 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 2 and d(1) = 2.

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

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A280875 Set a(1)=0, a(2)=1, a(3)=3; b(1)=1, b(2)=2; c(1)=3. Thereafter, a(n) is the smallest positive integer m such that m is not yet in sequence a, m-a(n-1) is not yet in sequence b, and m-a(n-2) is not yet in sequence c; set b(n-1)=m-a(n-1), c(n-2)=m-a(n-2).

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A257876 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 2.

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