A232639 -id:A232639 - cp's OEIS frontend

A232642 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S, and duplicates are deleted as they occur.

1, 2, 4, 3, 6, 5, 10, 8, 7, 14, 12, 11, 22, 9, 18, 16, 15, 30, 13, 26, 24, 23, 46, 20, 19, 38, 17, 34, 32, 31, 62, 28, 27, 54, 25, 50, 48, 47, 94, 21, 42, 40, 39, 78, 36, 35, 70, 33, 66, 64, 63, 126, 29, 58, 56, 55, 110, 52, 51, 102, 49, 98, 96, 95, 190, 44

Offset: 1

Clark Kimberling, Nov 28 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,4), g(3) = (3,6,5,10), etc. Concatenating these gives A232642, a permutation of the positive integers. For n > 1, the number of numbers in g(n) is 2*F(n+1), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x + 2 if 2*x + 2 has not already occurred.

Seen as triangle read by rows: A082560 with duplicates removed. - Reinhard Zumkeller, May 14 2015

			Each x begets x + 1 and 2*x + 2, but if either has already occurred it is deleted. Thus, 1 begets 2 and 4; then 2 begets 3 and 6, and 4 begets 5 and 10, so that g(3) = (3,6,5,10).
First 5 generations, also showing the places where duplicates were removed:
.  1:                                1
.  2:                2                               4
.  3:        3              6               5                10
.  4:    _       8      7       14      _       12       11       22
.  5:  _  __   9  18  _  16  15   30  _  __  13   26  __   24  23   46
These are the corresponding complete rows of triangle A082560:
.  1:                                1
.  2:                2                               4
.  3:        3              6               5                10
.  4:    4       8      7       14      6       12       11       22
.  5:  5  10   9  18  8  16  15   30  7  14  13   26  12   24  23   46

Clark Kimberling and Reinhard Zumkeller, Rows n = 1..17 of triangle, flattened, first 13 rows from Clark Kimberling
Index entries for sequences that are permutations of the natural numbers

Cf. A232559, A232639, A232643, A000045.

Cf. A128588 (row lengths), A033484 (right edges), A257956 (row sums), A082560.

import Data.List.Ordered (member); import Data.List (sort)
a232642 n k = a232642_tabf !! (n-1) !! (k-1)
a232642_row n = a232642_tabf !! (n-1)
a232642_tabf = f a082560_tabf [] where
   f (xs:xss) zs = ys : f xss (sort (ys ++ zs)) where
     ys = [v | v <- xs, not $ member v zs]
a232642_list = concat a232642_tabf
-- Reinhard Zumkeller, May 14 2015

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 2]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232642 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232643 *)

Keyword tabf added, to bring out function g, by Reinhard Zumkeller, May 14 2015

A232638 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x - 1 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 9, 8, 13, 11, 10, 17, 15, 14, 25, 12, 21, 19, 18, 33, 16, 29, 27, 26, 49, 23, 22, 41, 20, 37, 35, 34, 65, 31, 30, 57, 28, 53, 51, 50, 97, 24, 45, 43, 42, 81, 39, 38, 73, 36, 69, 67, 66, 129, 32, 61, 59, 58, 113, 55, 54, 105, 52, 101, 99

Offset: 1

Clark Kimberling, Nov 28 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x - 1 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2), g(3) = (3), g(4) = (4,5), etc. Concatenating these gives A232638, a permutation of the positive integers. For n > 1, the number of numbers in g(n) is F(n-1), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x - 1 if 2*x - 1 has not already occurred.

			Each x begets x + 1 and 2*x - 1, but if either has already occurred it is deleted. Thus, 1 begets 2, which begets 3, which begets 4 and 5, which beget 7 and (6,8), respectively.

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

Cf. A000045, A232559, A232639.

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] - 1]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232638 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232639 *)

A232640 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 1 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 9, 8, 15, 13, 12, 23, 10, 19, 17, 16, 31, 14, 27, 25, 24, 47, 21, 20, 39, 18, 35, 33, 32, 63, 29, 28, 55, 26, 51, 49, 48, 95, 22, 43, 41, 40, 79, 37, 36, 71, 34, 67, 65, 64, 127, 30, 59, 57, 56, 111, 53, 52, 103, 50, 99, 97, 96, 191

Offset: 1

Clark Kimberling, Nov 28 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 1 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,3), g(3) = (5,4,7), etc. Concatenating these gives A232640, a permutation of the positive integers. The number of numbers in g(n) is F(n), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x + 1 if 2*x + 1 has not already occurred.

			Each x begets x + 1 and 2*x + 1, but if either has already occurred it is deleted. Thus, 1 begets 2 and 3; then 2 begets only 5, and 3 begets (4,7), so that g(3) = (5,4,7).

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

Cf. A000045, A003754, A135533, A232559, A232639, A232641.

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 1]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* this sequence *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232641 *)

Conjecture: a(n) = A135533(A003754(n+1)) for n > 0. - Mikhail Kurkov, Feb 26 2023

A232641 Inverse permutation of the sequence of positive integers at A232640.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 9, 15, 8, 13, 12, 20, 11, 18, 17, 28, 16, 26, 25, 41, 14, 23, 22, 36, 21, 34, 33, 54, 19, 31, 30, 49, 29, 47, 46, 75, 27, 44, 43, 70, 42, 68, 67, 109, 24, 39, 38, 62, 37, 60, 59, 96, 35, 57, 56, 91, 55, 89, 88, 143, 32, 52, 51, 83

Offset: 1

Clark Kimberling, Nov 28 2013

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

Cf. A232559, A232639, A232640, A000045.

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 1]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]];
g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232640 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232641 *)

A232643 Inverse permutation of the sequence of positive integers at A232642.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 8, 14, 7, 12, 11, 19, 10, 17, 16, 27, 15, 25, 24, 40, 13, 22, 21, 35, 20, 33, 32, 53, 18, 30, 29, 48, 28, 46, 45, 74, 26, 43, 42, 69, 41, 67, 66, 108, 23, 38, 37, 61, 36, 59, 58, 95, 34, 56, 55, 90, 54, 88, 87, 142, 31, 51, 50, 82, 49

Offset: 1

Clark Kimberling, Nov 28 2013

Clark Kimberling and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Clark Kimberling)

Cf. A232559, A232639, A232642, A000045.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a232643 = (+ 1) . fromJust . (`elemIndex` a232642_list)
-- Reinhard Zumkeller, May 14 2015

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 2]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232642 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232643 *)

b-File corrected by Reinhard Zumkeller, May 14 2015

A232644 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 3 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 5, 3, 7, 6, 13, 4, 9, 8, 17, 15, 14, 29, 11, 10, 21, 19, 18, 37, 16, 33, 31, 30, 61, 12, 25, 23, 22, 45, 20, 41, 39, 38, 77, 35, 34, 69, 32, 65, 63, 62, 125, 27, 26, 53, 24, 49, 47, 46, 93, 43, 42, 85, 40, 81, 79, 78, 157, 36, 73, 71, 70, 141, 67, 66

Offset: 1

Clark Kimberling, Nov 28 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 3 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,5), g(3) = (3,7,6,13), etc. Concatenating these gives A232644, a permutation of the positive integers. For n > 2, the number of numbers in g(n) is L(n), where F = A000032, the Lucas numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x + 3 if 2*x + 3 has not already occurred.

			Each x begets x + 1 and 2*x + 3, but if either has already occurred it is deleted. Thus, 1 begets 2 and 5; then 2 begets 3 and 7, and 5 begets 6 and 13, so that g(3) = (3,7,6,13).

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

Cf. A000032, A232559, A232639, A232645.

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 3]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232644 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000032 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232645 *)

A232645 Inverse permutation of the sequence of positive integers at A232644.

Original entry on oeis.org

1, 2, 4, 8, 3, 6, 5, 10, 9, 16, 15, 26, 7, 13, 12, 21, 11, 19, 18, 31, 17, 29, 28, 47, 27, 45, 44, 73, 14, 24, 23, 39, 22, 37, 36, 60, 20, 34, 33, 55, 32, 53, 52, 86, 30, 50, 49, 81, 48, 79, 78, 128, 46, 76, 75, 123, 74, 121, 120, 196, 25, 42, 41, 68, 40, 66

Offset: 1

Clark Kimberling, Nov 28 2013

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

Cf. A232559, A232639, A232644.

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 3]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232644 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000032 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232645 *)

A232646 Sequence (or tree or triangle) generated by these rules: 1 is in S, and if x is in S, then 2x and 5x + 3 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 5, 4, 10, 25, 8, 20, 50, 125, 16, 40, 100, 250, 625, 32, 80, 200, 500, 1250, 3125, 64, 160, 400, 1000, 2500, 6250, 15625, 128, 320, 800, 2000, 5000, 12500, 31250, 78125, 256, 640, 1600, 4000, 10000, 25000, 62500, 156250, 390625, 512, 1280, 3200, 8000

Offset: 1

Clark Kimberling, Nov 28 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then 2*x and 5*x are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,5), g(3) = (4,10,25), etc. Concatenating these gives A232646, a permutation of the positive integers. For n > 2, the number of numbers in g(n) is n. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to 2*x if 2*x has not already occurred, and an edge from x to 3*x if 3*x has not already occurred.

			Each x begets 2*x and 5*x, but if either has already occurred it is deleted.  Thus, 1 begets 2 and 5; then 2 begets 4 and 10, and 5 begets only 25, so that g(3) = (4,10,25).  Writing generations as rows results in a triangle whose first five rows are as follows:
1
2 .... 5
4 .... 10 ... 25
8 .... 20 ... 50 ... 125
16 ... 40 ... 100 .. 250 .. 625

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

Cf. A232559, A232639, A036561.

x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, 2*x, 5*x}]]], {12}]; x  (* Peter J. C. Moses, Nov 27 2013 *)

Counting the top row as row 0 and writing for (2^i)*(5*j) , the numbers in row n are , , ..., <0,n>.

cp's OEIS Frontend

A232642 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S, and duplicates are deleted as they occur.

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A232638 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x - 1 are in S, and duplicates are deleted as they occur.

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A232640 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 1 are in S, and duplicates are deleted as they occur.

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A232641 Inverse permutation of the sequence of positive integers at A232640.

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Mathematica

A232643 Inverse permutation of the sequence of positive integers at A232642.

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Author

Keywords

Links

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Programs

Haskell

Mathematica

Extensions

A232644 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 3 are in S, and duplicates are deleted as they occur.

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Comments

Examples

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Mathematica

A232645 Inverse permutation of the sequence of positive integers at A232644.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A232646 Sequence (or tree or triangle) generated by these rules: 1 is in S, and if x is in S, then 2*x and 5*x + 3 are in S, and duplicates are deleted as they occur.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A232646 Sequence (or tree or triangle) generated by these rules: 1 is in S, and if x is in S, then 2x and 5x + 3 are in S, and duplicates are deleted as they occur.