A175498 -id:A175498 - cp's OEIS frontend

A230455 Lexicographically earliest sequence of distinct positive integers such that the absolute value of the first difference contains only distinct squares.

1, 2, 6, 15, 31, 56, 7, 43, 107, 26, 126, 5, 149, 318, 29, 225, 450, 9, 265, 589, 13, 374, 774, 45, 529, 1058, 34, 659, 1335, 39, 823, 1664, 64, 964, 3, 1092, 2248, 132, 1357, 2726, 22, 1466, 2987, 71, 1752, 3516, 35, 1884, 3820, 99, 2124, 4333, 108, 2412

Offset: 1

Paul Tek, Oct 19 2013

nonn

Is this a permutation of the positive integers ?

Will every square show up in the absolute value of the first difference ?

			The first differences of 1, 2, 6, 15, 31,... are 1^2, 2^2, 3^2, 4^2, 5^2, 7^2, 6^2,...

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, Perl program for this sequence

Cf. A081145, A175498, A230383, A230455.

Perl
```
See Link section.
```

A257877 Sequence (a(n)) generated by Rule 1 (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, 10, 18, 12, 21, 14, 24, 16, 7, 19, 30, 20, 33, 22, 36, 23, 38, 26, 42, 28, 13, 31, 48, 32, 51, 34, 54, 35, 17, 39, 60, 40, 63, 41, 65, 44, 69, 46, 72, 47, 74, 50, 78, 52, 25, 55, 27, 56, 87, 58, 90, 59, 29, 62, 96, 64, 99

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) = 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. A131388, A257915, A257705, A081145, A257883, A175498.

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 *)

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

A230383 Lexicographically earliest sequence of distinct positive integers such that the first difference contains only distinct positive or negative squares.

Original entry on oeis.org

1, 2, 6, 5, 14, 10, 26, 17, 42, 78, 29, 4, 53, 37, 101, 20, 120, 56, 137, 16, 160, 60, 24, 145, 314, 25, 221, 52, 277, 21, 310, 85, 341, 197, 521, 80, 441, 41, 482, 121, 605, 76, 476, 152, 681, 105, 730, 54, 630, 146, 822, 38, 767, 142, 926, 1767, 3, 903, 62

Offset: 1

Paul Tek, Oct 17 2013

nonn

Is this a permutation of the positive integers ?

Will every square show up in the first difference ?

			n  a(n)   First difference
-  ----   ----------------
1     1
          +1^2
2     2
          +2^2
3     6
          -1^2
4     5
          +3^2
5    14
          -2^2
6    10
          +4^2
7    26
          ...

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, Perl program for this sequence

Cf. A081145, A175498, A230455, A230560.

Perl
```
See Link section.
```

A257879 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 0.

Original entry on oeis.org

2, 1, 3, 4, 7, 5, 9, 6, 11, 17, 13, 8, 15, 23, 16, 10, 19, 29, 21, 12, 24, 14, 25, 38, 27, 41, 28, 43, 31, 47, 33, 18, 35, 53, 37, 20, 39, 59, 40, 22, 44, 65, 45, 68, 46, 70, 49, 26, 51, 77, 52, 79, 55, 83, 57, 30, 60, 32, 61, 92, 63, 95, 64, 34, 67, 101, 69

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

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

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

a[1] = 2; 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}]  (* A257879 *)
Table[d[k], {k, 1, zz}]  (* A257880 *)

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

A257882 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2.

Original entry on oeis.org

2, 1, 4, 5, 3, 7, 12, 9, 15, 11, 6, 13, 21, 14, 8, 17, 27, 19, 10, 22, 33, 23, 36, 25, 39, 26, 41, 29, 45, 31, 16, 34, 18, 35, 54, 37, 57, 38, 20, 42, 63, 43, 66, 44, 68, 47, 24, 49, 75, 51, 78, 53, 81, 55, 28, 58, 30, 59, 90, 61, 93, 62, 32, 65, 99, 67, 102

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

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

Cf. A257918, A257880, A257705, A081145, A257883, A175498.

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]];
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}]  (* A257882 *)
Table[d[k], {k, 1, zz}]  (* A257918 *)

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

A257884 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 1.

Original entry on oeis.org

0, 2, 1, 4, 8, 3, 9, 5, 10, 7, 14, 6, 15, 13, 21, 11, 22, 16, 26, 12, 24, 17, 30, 18, 32, 19, 34, 23, 39, 20, 37, 28, 46, 25, 44, 27, 47, 29, 50, 35, 57, 31, 54, 38, 62, 33, 58, 36, 63, 40, 66, 41, 69, 42, 71, 43, 73, 49, 80, 45, 77, 110, 48, 82, 51, 86, 52

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

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

Cf. A257883, A175498, A257705, A081145.

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 *)

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

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.

A327844 Table read by antidiagonals: the m-th row gives the sequence constructed by repeatedly choosing the smallest positive number not already in the row such that for each k = 1, ..., m, the k-th differences are distinct.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 4, 3, 6, 1, 2, 4, 3, 6, 10, 1, 2, 4, 3, 6, 11, 5, 1, 2, 4, 3, 6, 11, 5, 11, 1, 2, 4, 3, 6, 11, 5, 9, 7, 1, 2, 4, 3, 6, 11, 5, 9, 7, 12, 1, 2, 4, 3, 6, 11, 5, 9, 7, 13, 9, 1, 2, 4, 3, 6, 11, 5, 9, 7, 13, 10, 16, 1, 2, 4, 3

Offset: 1

Peter Kagey, Sep 29 2019

First row is A175498. Main diagonal is A327743.
The index of where the m-th row first differs from A327743 is 6, 15, 15, 16, 16, 194, 301, 301, 1036, 1036, 1036, 1037, ...
For example, T(6, 194) != A327743(194), but T(6, n) = A327743(n) for n < 194.

			Table begins:
1, 2, 4, 3, 6, 10, 5, 11, 7, 12,  9, 16, 8, 17, 15, 23, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 25, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 25, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 27, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 27, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 27, 14, ...

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A175498, A327743, A327845.

cp's OEIS Frontend

A230455 Lexicographically earliest sequence of distinct positive integers such that the absolute value of the first difference contains only distinct squares.

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A257877 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 3.

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A230383 Lexicographically earliest sequence of distinct positive integers such that the first difference contains only distinct positive or negative squares.

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Perl

A257879 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 0.

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Mathematica

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A257882 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2.

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A257884 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 1.

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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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Mathematica

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

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