A131388 -id:A131388 - cp's OEIS frontend

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

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.

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.

A257907 After 1, the first differences of A257906 (its d-sequence).

Original entry on oeis.org

1, 2, 3, -2, 4, -3, 5, -1, 7, -9, 8, -4, 9, -8, 10, -5, 15, -19, 11, -10, 12, -7, 17, -18, 16, -13, 19, -17, 21, -16, 26, -29, 23, -21, 25, -23, 27, -26, 28, -27, 29, -25, 33, -35, 31, -22, 40, -45, 35, -34, 36, -33, 39, -41, 37, -31, 43, -42, 44, -47, 41

Offset: 1

Clark Kimberling, May 16 2015

This is sequence (d(n)) generated by the following generic rule, "Rule 3" (see below) with parameters a(1) = 0 and d(1) = 1, while A257906 gives the corresponding sequence a(n).
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.
See A257905 for a guide to related sequences and conjectures.
An informal version of Rule 3 follows:  the sequences "d" and "a" are jointly generated, the "driving idea" idea being that each new term of "d" is obtained by a greedy algorithm.  See A131388 for a similar procedure (Rule 1).

			a(1) = 0, d(1) = 1;
a(2) = 2, d(2) = 2;
a(3) = 5, d(3) = 3;
a(4) = 3, d(4) = -2.

Clark Kimberling (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

After initial 1, the first differences of A257906 (the associated a-sequence for this rule).

Cf. A257905.

import Data.List ((\\))
a257907 n = a257907_list !! (n-1)
a257907_list = 1 : f [0] [1] where
   f xs@(x:_) ds = g [2 - x .. -1] where
     g [] = h : f ((x + h) : xs) (h : ds) where
                  (h:_) = [z | z <- [1..] \\ ds, x - z `notElem` xs]
     g (h:hs) | h `notElem` ds && y `notElem` xs = h : f (y:xs) (h:ds)
              | otherwise = g hs
              where y = x + h
-- Reinhard Zumkeller, Jun 03 2015

{a, f} = {{0}, {1}}; 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 *)

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.

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.

cp's OEIS Frontend

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

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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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A257907 After 1, the first differences of A257906 (its d-sequence).

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

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

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