A232559 -id:A232559 - cp's OEIS frontend

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

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 *)

A232798 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1, 3x - 1 and 3x + 1 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 11, 13, 8, 10, 6, 14, 16, 20, 22, 12, 32, 34, 38, 40, 9, 23, 25, 29, 31, 17, 19, 15, 41, 43, 47, 49, 21, 59, 61, 65, 67, 35, 37, 33, 95, 97, 101, 103, 39, 113, 115, 119, 121, 26, 28, 24, 68, 70, 74, 76, 30, 86, 88, 92, 94, 18, 50, 52, 56

Offset: 1

Clark Kimberling, Nov 30 2013

Let S be the sequence (or tree) of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1, 3*x - 1, and 3*x + 1 are in S. Then S is a permutation of the positive integers. Deleting duplicates as they occur, the generations of S are given by g(1) = (1), g(2) = (2,4), g(3) = (3,5,7,11,13), etc. Concatenating these gives A232798. The position of n in S gives the inverse permutation of S, as in A232799.

			Each x begets x + 1, 3*x - 1 and 3*x + 1, but if any of these has already occurred it is deleted.  Thus, 1 begets (2,4); then 2 begets (3,5,7) and 4 begets (11,13), making g(3) = (3,5,7,11,13), etc.

Clark Kimberling, Table of n, a(n) for n = 1..4000

Cf. A232559, A232799.

x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, 3 x - 1, 3 x + 1}]]], {8}];  x (* A232798 *)
y = Flatten[Table[Position[x, n], {n, 1, 100}]] (* A232799 *)

A232799 Inverse permutation of the sequence of positive integers at A232798.

Original entry on oeis.org

1, 2, 4, 3, 5, 11, 6, 9, 21, 10, 7, 16, 8, 12, 28, 13, 26, 62, 27, 14, 33, 15, 22, 52, 23, 50, 120, 51, 24, 57, 25, 17, 40, 18, 38, 91, 39, 19, 45, 20, 29, 69, 30, 67, 161, 68, 31, 74, 32, 63, 151, 64, 149, 359, 150, 65, 156, 66, 34, 81, 35, 79, 190, 80, 36

Offset: 1

Clark Kimberling, Nov 30 2013

Clark Kimberling, Table of n, a(n) for n = 1..300

Cf. A232559, A232798.

x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, 3 x - 1, 3 x + 1}]]], {8}];  x (* A232798 *)
y = Flatten[Table[Position[x, n], {n, 1, 157}]] (* A232799 *)

A232890 Denominator of n-th term of sequence (or tree) S of all rational numbers generated by these rules: 0 is in S; if x is in S then x + 1 is in S, and if x + 1 is nonzero, then -1/(x + 1) is in S; duplicates are deleted as they occur.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 2, 3, 1, 5, 4, 3, 3, 5, 2, 5, 3, 1, 1, 6, 5, 4, 4, 7, 3, 8, 5, 2, 2, 7, 5, 3, 3, 4, 1, 7, 6, 5, 5, 9, 4, 11, 7, 3, 3, 11, 8, 5, 5, 7, 2, 9, 7, 5, 5, 8, 3, 7, 4, 1, 1, 8, 7, 6, 6, 11, 5, 14, 9, 4, 4, 15, 11, 7, 7, 10, 3

Offset: 1

Clark Kimberling, Dec 02 2013

Let S be the sequence (or tree) of numbers generated by these rules: 0 is in S; if x is in S then x + 1 is in S, and if x + 1 is nonzero, then -1/(x + 1) is in S. Deleting duplicates as they occur, the generations of S are given by g(1) = (0), g(2) = (1,-1), g(3) = (2,-1/2), g(4) = (3, -1/3, 1/2, -2), ... Concatenating gives 0, 1, -1, 2, -1/2, 3, -1/3, 1/2, -2, 4, -1/4, ...

Conjectures: If b/c is a positive rational number, the position of n + b/c for n >= 0 forms a linear recurrence sequence with signature (1,1), and the position of -n - b/c forms a linear recurrence sequence with signature (4, -4, 1). For n>=1, the numbers -(1 + 1/n) are terminal nodes in the tree, and their positions are linearly recurrent with signature (2,0,-1). For n >=3, the n-th generation g(n) consists of F(n-1) positive numbers and F(n-1) negative numbers, where F = A000045, the Fibonacci numbers.

			To generate S, the number 0 begets (1,-1), whence 1 begets 2 and -1/2, whereas -1 begets 0 and -1/2, both of which are (deleted )duplicates, so that g(3) = (2, -1/2).  The resulting concatenation of all the generations g(n) begins with 0, 1, -1, 2, -1/2, 3, -1/3, 1/2, -2, 4, -1/4, so that A232890 begins with 1,1,1,1,2,1,3,2,1,1,4.

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

Cf. A232559, A232868, A000045.

Off[Power::infy]; x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, -1/(x + 1)} /. ComplexInfinity -> 0]]], {8}]; x
On[Power::infy]; Denominator[x]  (* Peter J. C. Moses, Nov 29 2013 *)

A232895 Sequence (or tree) S of all positive integers in the order generated by these rules: 1 and 2 are in S; if x is in S then x + 2 and 2*x are in S, where duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 12, 16, 9, 14, 20, 24, 18, 32, 11, 28, 22, 40, 26, 48, 36, 34, 64, 13, 30, 56, 44, 42, 80, 52, 50, 96, 38, 72, 68, 66, 128, 15, 60, 58, 112, 46, 88, 84, 82, 160, 54, 104, 100, 98, 192, 76, 74, 144, 70, 136, 132, 130, 256, 17, 62

Offset: 1

Clark Kimberling, Dec 02 2013

Let S be the sequence (or tree) of numbers generated by these rules: 1 and 2 are in S; if x is in S then x + 2 and 2*x are in S, where duplicates are deleted as they occur. Every positive integer occurs exactly once in S, so that S is a permutation of the natural numbers. Deleting duplicates as they occur, the generations of S are given by g(1) = (1,2), g(2) = (3,4), g(3) = (5,6,8), g(4) = (7,10,12,16), ... Concatenating gives 1,2,3,4,5,6,8,... Conjecture: the position of the n-th odd positive integer in S is the linearly recurrent sequence given by A232896(n) for n>=1.

			To generate S, start with g(1) = (1,2).  Then 1 begets 3 and 2, but 2 is deleted as a duplicate, and 2 begets 4 and 4, of which the second 4 is deleted; thus g(2) = (3,4).

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

Cf. A232559, A232896.

x = {1, 2}; dx = 0; Do[x = DeleteDuplicates[Flatten[AppendTo[x, Transpose[{# + 2, 2*#}] &[Drop[x, Length[x] - dx]]]]]; dx = Length[x] - dx, {31}]; x  (* A232895 *)
t = Flatten[Position[Denominator[x/2], 2]] (* A232896 conjectured *)
(* Peter J. C. Moses, Dec 02 2013 *)

A233559 Tree read by levels generated by these rules: 1 is at the top, and the node x has child nodes x+1, 2x, and 3x, where duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 8, 12, 7, 18, 10, 27, 15, 16, 24, 13, 36, 14, 21, 19, 54, 11, 20, 30, 28, 81, 45, 17, 32, 48, 25, 72, 26, 39, 37, 108, 42, 22, 63, 38, 57, 55, 162, 33, 40, 60, 31, 90, 29, 56, 84, 82, 243, 46, 135, 34, 51, 64, 96, 49, 144, 50, 75, 73

Offset: 1

Clark Kimberling, Dec 14 2013

Every positive integer occurs exactly once in S, so that S is a permutation of the natural numbers. Deleting duplicates as they occur, the generations of S are given by g(1) = (1), g(2) = (2,3), g(3) = (4,6,9), g(4) = (5,8,12,7,18,10,27), ... Concatenating gives 1,2,3,4,6,9,5,...

			To generate S, start with g(1) = (1). Then 1 begets 2 and 3; then 2 begets 4 and 3 begets 6 and 9.

Clark Kimberling, Table of n, a(n) for n = 1..5000

Cf. A232559, A233560.

x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, 2 x, 3 x}]]], {8}];  x (* A233559 *)
y = Flatten[Table[Position[x, n], {n, 1, 157}]] (* A233560 *)

Name edited by Ivan Neretin, Nov 26 2016

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>.

A274455 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x -1 and 2*x are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 0, 2, -1, 4, -2, 3, 8, -3, -4, 6, 7, 16, -6, -5, -8, 5, 12, 14, 15, 32, -7, -12, -10, -9, -16, 10, 11, 24, 13, 28, 30, 31, 64, -14, -13, -24, -11, -20, -18, -17, -32, 9, 20, 22, 23, 48, 26, 27, 56, 29, 60, 62, 63, 128, -15, -28, -26, -25, -48, -22, -21

Offset: 1

Clark Kimberling, Jun 23 2016

Every integer occurs exactly once. The rules for this tree become identical to those for A232559 when "x + 1" is substituted for "x - 1".

For n > 3, the n-th generation has F(n) nodes, of which F(n-1) are positive and F(n-2) are negative, where F = A000045, the Fibonacci numbers.

			Generation g(1) consists of the seed, 1; generation g(2) consists of 0 and 2 from which 0 begets -1 and 0, but this 0 is a duplicate and is removed, while 2 begets 1 and 4, with 1 removed, so that g(3) = {-1,4}.  Thereafter, g(4) = {-2,3,8}, g(5) = {-3,-4,6,7,16}, etc.

Clark Kimberling, Table of n, a(n) for n = 1..10946

Cf. A000045, A232559.

z = 12; g[1] = {1}; g[2] = {0, 2};
g[n_] := Riffle[g[n - 1] - 1, 2 g[n - 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}]]  (*A274455*)

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

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

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A232798 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1, 3*x - 1 and 3*x + 1 are in S, and duplicates are deleted as they occur.

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A232799 Inverse permutation of the sequence of positive integers at A232798.

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A232890 Denominator of n-th term of sequence (or tree) S of all rational numbers generated by these rules: 0 is in S; if x is in S then x + 1 is in S, and if x + 1 is nonzero, then -1/(x + 1) is in S; duplicates are deleted as they occur.

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A232895 Sequence (or tree) S of all positive integers in the order generated by these rules: 1 and 2 are in S; if x is in S then x + 2 and 2*x are in S, where duplicates are deleted as they occur.

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A233559 Tree read by levels generated by these rules: 1 is at the top, and the node x has child nodes x+1, 2*x, and 3*x, where duplicates are deleted as they occur.

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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.

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A232798 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1, 3x - 1 and 3x + 1 are in S, and duplicates are deleted as they occur.

A233559 Tree read by levels generated by these rules: 1 is at the top, and the node x has child nodes x+1, 2x, and 3x, where duplicates are deleted as they occur.

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.