A257706 -id:A257706 - cp's OEIS frontend

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

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.

A257698 Number of steps from n to 1 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = 3/2.

Original entry on oeis.org

0, 1, 3, 2, 6, 3, 5, 7, 13, 4, 6, 8, 12, 14, 16, 5, 13, 9, 11, 13, 17, 15, 17, 6, 12, 14, 16, 10, 14, 14, 16, 18, 28, 16, 18, 7, 11, 13, 15, 15, 21, 11, 13, 15, 19, 15, 17, 19, 27, 29, 31, 17, 21, 8, 10, 12, 18, 14, 16, 16, 20, 22, 24, 12, 24, 16, 18, 20, 24

Offset: 1

Clark Kimberling, May 04 2015

			5->7->10->6->4->2->1, total of 6 steps, so that a(5) = 6.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A257699-A257706.

r = 3/2; f[x_] := If[OddQ[x], Floor[r *x], Floor[x/r]]
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1 + Length[g[n]], {n, 1, 200}]

A257699 Number of steps from n to 0 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = 5/2.

Original entry on oeis.org

2, 1, 7, 3, 5, 2, 6, 8, 10, 4, 14, 4, 6, 6, 10, 3, 5, 7, 13, 9, 11, 9, 13, 11, 13, 5, 13, 15, 17, 5, 9, 5, 7, 7, 13, 7, 9, 11, 15, 4, 6, 4, 18, 6, 8, 8, 12, 14, 16, 10, 16, 10, 12, 12, 16, 10, 12, 14, 22, 12, 14, 12, 16, 14, 16, 6, 12, 14, 16, 16, 20, 16, 18

Offset: 1

Clark Kimberling, May 04 2015

			5->12->4->1->2->0, total of 5 steps, so that a(5) = 5.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A257698, A257700-A257706.

r = 5/2; f[x_] := If[OddQ[x], Floor[r *x], Floor[x/r]]
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1 + Length[g[n]], {n, 1, 200}]

A257700 Number of steps from n to 1 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = 5/3.

Original entry on oeis.org

0, 1, 5, 2, 4, 6, 10, 3, 11, 7, 9, 11, 17, 4, 10, 12, 14, 8, 26, 12, 16, 18, 20, 5, 9, 11, 19, 13, 15, 9, 25, 27, 37, 13, 15, 17, 23, 19, 23, 6, 8, 10, 14, 12, 18, 20, 22, 14, 32, 10, 24, 26, 28, 28, 36, 38, 42, 14, 16, 18, 22, 24, 34, 20, 22, 24, 30, 7, 13

Offset: 1

Clark Kimberling, May 04 2015

			5->8->4->2->1, total of 4 steps, so that a(5) = 4.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A257698, A257699, A257701-A257706.

r = 5/3; f[x_] := If[OddQ[x], Floor[r *x], Floor[x/r]]
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1 + Length[g[n]], {n, 1, 200}]

A257701 Number of steps from n to 1 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = sqrt(3).

Original entry on oeis.org

0, 1, 5, 2, 4, 6, 8, 3, 16, 5, 9, 7, 9, 4, 15, 17, 21, 6, 8, 10, 12, 8, 18, 10, 14, 16, 18, 18, 20, 22, 32, 7, 11, 9, 11, 11, 13, 13, 17, 19, 23, 11, 13, 15, 19, 17, 21, 19, 21, 19, 21, 23, 31, 33, 37, 8, 10, 12, 14, 10, 22, 12, 16, 12, 14, 14, 16, 18, 22

Offset: 1

Clark Kimberling, May 04 2015

			7->12->6->3->5->8->4->2->1, total of 8 steps, so that a(7) = 8.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A257698-A257700, A257702-A257706.

r = Sqrt[3]; f[x_] := If[OddQ[x], Floor[r *x], Floor[x/r]]
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1 + Length[g[n]], {n, 1, 200}]

A257702 Number of steps from n to 0 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = sqrt(5).

Original entry on oeis.org

2, 1, 3, 3, 7, 2, 12, 4, 6, 4, 6, 8, 12, 3, 11, 13, 15, 5, 7, 5, 7, 7, 11, 5, 9, 7, 9, 9, 11, 13, 17, 4, 10, 12, 14, 14, 16, 14, 18, 16, 24, 6, 8, 8, 10, 6, 10, 8, 16, 8, 10, 12, 14, 6, 8, 10, 16, 10, 14, 8, 10, 10, 12, 10, 14, 12, 18, 14, 16, 18, 20, 5, 9

Offset: 1

Clark Kimberling, May 04 2015

			5->11->24->10->4->1->2->0, total of 7 steps, so that a(5) = 7.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A257698-A257701, A257703-A257706.

r = Sqrt[5]; f[x_] := If[OddQ[x], Floor[r *x], Floor[x/r]]
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1 + Length[g[n]], {n, 1, 200}]

A257703 Number of steps from n to 0 using this algorithm: x -> floor(e*x) if x is odd, and x -> floor(x/e) if x is even.

Original entry on oeis.org

2, 1, 3, 3, 11, 2, 8, 2, 4, 4, 8, 4, 10, 12, 14, 12, 14, 3, 7, 9, 13, 3, 5, 3, 7, 5, 9, 5, 7, 9, 11, 9, 17, 5, 9, 11, 13, 11, 13, 13, 19, 15, 17, 13, 15, 13, 25, 15, 23, 4, 6, 8, 10, 8, 14, 10, 12, 14, 16, 4, 8, 4, 8, 6, 8, 4, 6, 8, 12, 8, 10, 6, 8, 10, 20

Offset: 1

Clark Kimberling, May 04 2015

			5->13->35->95->258->94->34->12->4->1->2->0, total of 11 steps, so that a(5) = 11.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A257698-A257702, A257704-A257706.

r = E; f[x_] := If[OddQ[x], Floor[r *x], Floor[x/r]]
g[x_] := Drop[FixedPointList[f, x], -1];
Table[-1 + Length[g[n]], {n, 1, 200}]

cp's OEIS Frontend

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

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A257698 Number of steps from n to 1 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = 3/2.

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A257699 Number of steps from n to 0 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = 5/2.

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A257700 Number of steps from n to 1 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = 5/3.

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A257701 Number of steps from n to 1 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = sqrt(3).

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A257702 Number of steps from n to 0 using this algorithm: x -> floor(r*x) if x is odd, and x -> floor(x/r) if x is even, where r = sqrt(5).

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A257703 Number of steps from n to 0 using this algorithm: x -> floor(e*x) if x is odd, and x -> floor(x/e) if x is even.

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