A183079 -id:A183079 - cp's OEIS frontend

A220348 Index of row where n occurs in A183079.

1, 2, 3, 3, 4, 4, 4, 5, 5, 4, 5, 6, 6, 5, 5, 6, 7, 7, 6, 6, 5, 7, 8, 8, 7, 7, 6, 5, 8, 9, 9, 8, 8, 7, 6, 6, 9, 10, 10, 9, 9, 8, 7, 7, 6, 10, 11, 11, 10, 10, 9, 8, 8, 7, 5, 11, 12, 12, 11, 11, 10, 9, 9, 8, 6, 6, 12, 13, 13, 12, 12, 11, 10, 10, 9, 7, 7, 7, 13

Offset: 1

Reinhard Zumkeller, Dec 12 2012

nonn

A183079 seen as flattened sequence is a permutation of the natural numbers, therefore for each n there exists exactly 1 row in A183079 containing n.

In this sequence each n >= 2 occurs a total of 2^(n-2) times. - Antti Karttunen, May 18 2015

Reinhard Zumkeller (first 250 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10440

Cf. A029837, A070941, A183079, A220347.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a220348 n = fromJust (findIndex (elem n) a183079_tabf) + 1

(* b is A220347 *) b[n_] := b[n] = With[{r = (-1 + Sqrt[8n + 1])/2}, Which[n <= 1, n, IntegerQ[r], 2b[Floor[Sqrt[2n] + 1/2]] - 1, True, 2b[n - Floor[r]]]];
a[n_] := 1 + IntegerLength[b[n] - 1, 2];
Array[a, 100] (* Jean-François Alcover, Dec 05 2021 *)

(define (A220348 n) (+ 1 (A029837 (A220347 n))))
;; Antti Karttunen, May 18 2015

(define (A220348 n) (A070941 (+ -1 (A220347 n))))
;; Antti Karttunen, May 18 2015

a(n) = 1 + A029837(A220347(n)) = A070941(A220347(n)-1). - Antti Karttunen, May 18 2015

Name edited by Michel Marcus, Jan 26 2022

A183209 Tree generated by floor(3n/2): a(1) = 1, a(2n) = (3a(n))-1, a(2n+1) = floor((3a(n+1))/2).

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 7, 14, 6, 11, 12, 23, 10, 20, 21, 41, 9, 17, 16, 32, 18, 35, 34, 68, 15, 29, 30, 59, 31, 62, 61, 122, 13, 26, 25, 50, 24, 47, 48, 95, 27, 53, 52, 104, 51, 101, 102, 203, 22, 44, 43, 86, 45, 89, 88, 176, 46, 92, 93, 185, 91, 182, 183, 365, 19, 38, 39, 77, 37

Offset: 1

Clark Kimberling, Dec 30 2010

A permutation of the positive integers. See the comment at A183079. Leftmost branch of tree is essentially A061418. Rightmost: A007051.

			First levels of the tree:
                      1
                      2
            3                   5
          4   8               7   14

Cf. A183079, A007051, A016789, A183207, A183208, A032766, A191450.
Similar permutations: A048673, A254103.
Inverse permutation: A259431.

import Data.List (transpose)
a183209 n k = a183209_tabf !! (n-1) !! (k-1)
a183209_row n = a183209_tabf !! (n-1)
a183209_tabf = [1] : iterate (\xs -> concat $
   transpose [map a032766 xs, map (a016789 . subtract 1) xs]) [2]
a183209_list = concat a183209_tabf
-- Reinhard Zumkeller, Jun 27 2015

f:= proc(n) option remember;
  if n::even then 3*procname(n/2)-1
  else floor(3*procname((n+1)/2)/2)
  fi
end proc:
f(1):= 1:
seq(f(n), n=1..100); # Robert Israel, Jan 26 2015

a[1]=1; a[n_] := a[n] = If[EvenQ[n], 3a[n/2]-1, Floor[3a[(n+1)/2]/2] ]; Array[a, 100] (* Jean-François Alcover, Feb 02 2018 *)

def a(n):
    if n==1: return 1
    if n%2==0: return 3*a(n//2) - 1
    else: return (3*a((n - 1)//2 + 1))//2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

Let L(n)=floor(3n/2).
Let U(n)=3n-1.  U is the complement of L.
The tree-array T(n,k) is then given by rows:
T(0,0)=1; T(1,0)=2;
T(n,2j)=L(T(n-1),j);
T(n,2j+1)=U(T(n-1),j);
for j=0,1,...,2^(n-1)-1, n>=2.
From Antti Karttunen, Jan 26 2015: (Start)
a(1) = 1, a(2n) = (3*a(n))-1, a(2n+1) = A032766(a(n+1)) = floor((3*a(n+1))/2).
Other identities:
a(2^n) = A007051(n) for all n >= 0. [A property shared with A048673 and A254103.]
(End)

Formula to the name-field added by Antti Karttunen, Jan 26 2015

A183089 Tree generated by the lucky numbers: a(1) = 1; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n+1)), where lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 6, 21, 11, 13, 8, 31, 14, 15, 10, 87, 29, 37, 17, 49, 19, 25, 12, 141, 42, 51, 20, 63, 22, 33, 16, 517, 112, 133, 40, 189, 50, 69, 24, 259, 64, 75, 27, 111, 35, 43, 18, 925, 177, 211, 56, 267, 66, 79, 28, 339, 83, 93, 30, 159, 45, 67, 23, 4129, 618, 685, 143, 855, 167, 201, 54, 1275, 234, 261, 65, 391, 90, 105, 34

Offset: 1

Clark Kimberling, Dec 24 2010

A permutation of the positive integers. See the comment at A183079.

			Top 6 levels of the binary tree:
                                     1
                                     |
                  ...................2...................
                 3                                       4
       7......../ \........5                   9......../ \........6
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  21       11         13       8          31       14         15       10
87  29   37  17     49  19   25 12     141  42   51  20     63  22   33  16
...
From the level 3 to the level 4: 3 --> (7,5) and 4 --> (9,6).

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A257690.
Cf. A000959, A050505.
Cf. A257726 (similar permutation with a slightly different definition, resulting the first differing term at n=9, where a(9) = 21, while A257726(9) = 13), A257735 - A257738.
Cf. A183079, A237739 (other similar permutations).

Let L(n) = A000959(n), the n-th lucky number.
Let U(n) = A050505(n), the n-th unlucky numbers.
The tree-array T(n,k) is then given by rows:
T(0,0) = 1; T(1,0) = 2;
T(n,2j) = L(T(n-1),j);
T(n,2j+1) = U(T(n-1),j);
for j = 0, 1, ..., 2^(n-1) - 1, n >= 2.
a(1) = 1; a(2n) = A050505(a(n)), a(2n+1) = A000959(a(n+1)). - Antti Karttunen, May 09 2015

Added a formula to the Name field and more terms, edited Example section - Antti Karttunen, May 09 2015

A183231 First of two complementary trees generated by the triangular numbers. The second tree is A183232.

Original entry on oeis.org

1, 4, 3, 19, 7, 13, 6, 229, 25, 43, 11, 118, 18, 34, 10, 26794, 250, 376, 32, 1033, 52, 89, 16, 7258, 133, 208, 24, 664, 42, 76, 15, 359026204, 27025, 31876, 272, 71629, 403, 593, 40, 536128, 1078, 1483, 62, 4184, 102, 169, 22, 26357428

Offset: 1

Clark Kimberling, Jan 02 2011

Begin with the main tree A183079 generated by the triangular numbers:
......................1
......................2
.............3.................4
.........6.......5........10........7
.......21..9...15..8....55..14....28..11
Every n>2 is in the subtree from 3 or the subtree from 4.
Therefore, on subtracting 2 from all entries of those subtrees, we obtain complementary trees:  A183231 and A183232.

			First three levels:
............1
.......4.........3
....19...7.....13..6

Cf. A183079, A183232 (second tree), A183233.

See the formulas at A183079 and A183233.

A220347 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = (2a(n))-1, a(nontriangular(n)) = 2n, where triangular = A000217, nontriangular = A014132.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 16, 24, 20, 14, 11, 32, 48, 40, 28, 22, 9, 64, 96, 80, 56, 44, 18, 15, 128, 192, 160, 112, 88, 36, 30, 23, 256, 384, 320, 224, 176, 72, 60, 46, 19, 512, 768, 640, 448, 352, 144, 120, 92, 38, 13, 1024, 1536, 1280, 896, 704, 288

Offset: 1

Reinhard Zumkeller, Dec 12 2012

nonn

Inverse permutation of A183079, when seen as a flattened sequence.

Reinhard Zumkeller (first 250 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10440
Index entries for sequences that are permutations of the natural numbers

Inverse: A183079.
Cf. A000217, A002024, A010054, A014132, A083920, A220348.
Cf. also a similar permutation A257797 from which this differs for the first time at n=15, where a(15) = 11, while A257797(15) = 9.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a220347 =  (+ 1) . fromJust . (`elemIndex` a183079_list)

a[n_] := a[n] = With[{r = (-1 + Sqrt[8n + 1])/2}, Which[n <= 1, n, IntegerQ[r], 2 a[Floor[Sqrt[2n] + 1/2]] - 1, True, 2 a[n - Floor[r]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 05 2021 *)

;; With memoizing definec-macro.
(definec (A220347 n) (cond ((<= n 1) n) ((zero? (A010054 n)) (* 2 (A220347 (A083920 n)))) (else (+ -1 (* 2 (A220347 (A002024 n)))))))
;; Antti Karttunen, May 18 2015

a(1) = 1; for n > 1: if A010054(n) = 1 [i.e., if n is triangular], then a(n) = (2*a(A002024(n)))-1, otherwise a(n) = 2*a(A083920(n)). - Antti Karttunen, May 18 2015

Old name moved to comments by Antti Karttunen, May 18 2015

A183169 Tree generated by the squares.

Original entry on oeis.org

1, 2, 4, 3, 16, 6, 9, 5, 256, 20, 36, 8, 81, 12, 25, 7, 65536, 272, 400, 24, 1296, 42, 64, 11, 6561, 90, 144, 15, 625, 30, 49, 10, 4294967296, 65792, 73984, 288, 160000, 420, 576, 29, 1679616, 1332, 1764, 48, 4096, 72, 121, 14

Offset: 1

Clark Kimberling, Dec 28 2010

A permutation of the positive integers. See the comment at A183079 (tree generated by the triangular numbers). The leftmost numbers (1,2,4,16,...) are, after the initial 1, given by A001146. The rightmost numbers (1,2,3,5,7,10,...) are, after the initial 1, the iterates of the nonsquare function; see a comment at A033638.

			First levels of the tree:
......................1
......................2
...........4.....................3
.......16.......6...........9..........5
...256...20...36..8......81...12....25...7

Cf. A183079, A000037, A183169, A001146, A033638.

Let L(n) be the n-th square (A000290).
Let U(n) be the n-th nonsquare (A000037).
The tree-array T(n,k) is then given by rows:
T(0,0)=1; T(1,0)=2;
T(n,2j)=L(T(n-1),j);
T(n,2j+1)=U(T(n-1),j);
for j=0,1,...,2^(n-1)-1, n>=2.

A183232 Second of two complementary trees generated by the triangular numbers. The other tree is A183231.

Original entry on oeis.org

2, 8, 5, 53, 12, 26, 9, 1538, 63, 103, 17, 404, 33, 64, 14, 1186568, 1593, 2143, 74, 5563, 117, 188, 23, 82619, 432, 628, 41, 2209, 75, 134, 20, 703974775733, 1188108, 1272808, 1649, 2301583, 2208, 2924, 86, 15487393

Offset: 1

Clark Kimberling, Jan 02 2011

See A183231 (first tree).

			First 3 levels:
....................2
...............8...........5
............53...12.....26...9

Cf. A183079, A183231, A183244.

See the formulas at A183231 and A183244.

A183542 First of two complementary trees generated by the Wythoff sequences.

Original entry on oeis.org

1, 2, 5, 4, 8, 9, 16, 7, 13, 14, 24, 15, 26, 27, 45, 12, 21, 22, 37, 23, 39, 40, 66, 25, 42, 43, 71, 44, 73, 74, 121, 20, 34, 55, 58, 36, 60, 61, 100, 38, 63, 64, 105, 65, 107, 108, 176, 41, 68, 69, 113, 70, 115, 116, 189, 72, 118, 119, 194, 120, 196, 197, 320

Offset: 1

Clark Kimberling, Jan 05 2011

Begin with the main tree A074049 generated by the Wythoff sequences:
...................1
...................2
...........3.................5
.......4.......7........8........13
.....6..10...11..18....12..20...21..34
Every n >2 is in the subtree from 3 or the subtree from 5.  Therefore, on subtracting 2 from all entries in those subtrees, we obtain complementary trees: A183342 and A183543.

			First three levels:
...................1
.............2............3
..........4.....8......9.....16

Cf. A074049, A183543, A183544,

A183079 (definition of tree generated by a sequence).

See the formulas at A074049 and A183544.

A257798 Permutation of natural numbers: a(1) = 1; a(2n) = nontriangular(a(n)), a(2n+1) = triangular(1+a(n)), where triangular = A000217, nontriangular = A014132.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 7, 15, 9, 28, 8, 21, 14, 66, 11, 36, 20, 136, 13, 55, 35, 435, 12, 45, 27, 253, 19, 120, 77, 2278, 16, 78, 44, 703, 26, 231, 152, 9453, 18, 105, 65, 1596, 43, 666, 464, 95266, 17, 91, 54, 1081, 34, 406, 275, 32385, 25, 210, 135, 7381, 89, 3081, 2345, 2598060, 22, 153, 90, 3160, 53, 1035, 740, 248160, 33

Offset: 1

Antti Karttunen, May 18 2015

nonn

This sequence can be represented as a binary tree. Each left hand child is produced as A014132(n) and each right hand child as A000217(1+n), when a parent contains n >= 1:
                                    1
                 ................../ \..................
                2                                       3
      4......../ \........6                   5......../ \.......10
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       15          9       28          8       21         14       66
11 36   20  136     13 55   35  435     12 45   27  253    19  120  77 2278
etc.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A257797.
Cf. A000217, A014132.
Cf. also permutation A183079.

a(1)=1; after which: a(2n) = A014132(a(n)), a(2n+1) = A000217(a(n)+1).

A183080 Tree generated by the Beatty sequence of 3-sqrt(2).

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 7, 13, 6, 10, 12, 21, 11, 18, 20, 35, 9, 16, 15, 27, 19, 32, 33, 56, 17, 29, 28, 48, 31, 54, 55, 94, 14, 24, 25, 43, 23, 40, 42, 73, 30, 51, 50, 86, 52, 89, 88, 151, 26, 46, 45, 78, 44, 75, 76, 129, 49, 83, 85, 146, 87, 148, 149, 254

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers. See the note at A183079.

			First five rows:
1
2
3 5
4 8 7 13
6 10 12 21 11 18 20 35

Cf. A183079, A178528, A074049.

a = {1, 2}; row = {a[[-1]]}; r = 3 - Sqrt[2]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)

Let L(n)=floor(n*r), U(n)=floor(n*s), where r=3-sqrt(2) and s=r/(r-1).

The tree-array T(n,k) is then given by rows: T(0,0) = 1; T(1,0) = 2; T(n,2j) = L(T(n-1),j); T(n,2j+1) = U(T(n-1),j); for j=0,1,...,2^(n-1)-1, n>=2.

cp's OEIS Frontend

A220348 Index of row where n occurs in A183079.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Scheme

Scheme

Formula

Extensions

A183209 Tree generated by floor(3n/2): a(1) = 1, a(2n) = (3*a(n))-1, a(2n+1) = floor((3*a(n+1))/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

Extensions

A183089 Tree generated by the lucky numbers: a(1) = 1; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n+1)), where lucky = A000959, unlucky = A050505.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A183231 First of two complementary trees generated by the triangular numbers. The second tree is A183232.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A220347 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = (2*a(n))-1, a(nontriangular(n)) = 2*n, where triangular = A000217, nontriangular = A014132.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Scheme

Formula

Extensions

A183169 Tree generated by the squares.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A183232 Second of two complementary trees generated by the triangular numbers. The other tree is A183231.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A183542 First of two complementary trees generated by the Wythoff sequences.

Views

Author

A183209 Tree generated by floor(3n/2): a(1) = 1, a(2n) = (3a(n))-1, a(2n+1) = floor((3a(n+1))/2).

A220347 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = (2a(n))-1, a(nontriangular(n)) = 2n, where triangular = A000217, nontriangular = A014132.