A220347 -id:A220347 - cp's OEIS frontend

A183079 Tree generated by the triangular numbers: a(1) = 1; a(2n) = nontriangular(a(n)), a(2n+1) = triangular(a(n+1)), where triangular = A000217, nontriangular = A014132.

1, 2, 3, 4, 6, 5, 10, 7, 21, 9, 15, 8, 55, 14, 28, 11, 231, 27, 45, 13, 120, 20, 36, 12, 1540, 65, 105, 19, 406, 35, 66, 16, 26796, 252, 378, 34, 1035, 54, 91, 18, 7260, 135, 210, 26, 666, 44, 78, 17, 1186570, 1595, 2145, 76, 5565, 119, 190, 25, 82621, 434

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers.
In general, suppose that L and U are complementary sequences of positive integers such that
(1) L(1)=1; and
(2) if n>1, then n=L(k) or n=U(k) for some kThe tree generated by the sequence L is defined as follows:
  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.
The numbers, taken in the order generated, form a permutation of the positive integers.

			First levels of the tree:
                                    1
                                    |
                 ...................2...................
                3                                       4
      6......../ \........5                   10......./ \........7
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  21      9          15       8          55       14         28      11
231 27  45 13     120  20   36 12    1540  65  105  19    406  35  66  16
Beginning with 3 and 4, the numbers are generated in pairs, such as (3,4), (6,5), (10,7), (21,9),...
In all such pairs, the first number belongs to A000217; the second, to A014132.

Reinhard Zumkeller, Rows n = 1..14 of triangle, flattened
Index entries for sequences that are permutations of the natural numbers

Cf. A000217, A014132, A074049.
Cf. A220347 (inverse), A220348.
Cf. A183089, A183209 (similar permutations), also A257798.

a183079 n k = a183079_tabf !! (n-1) !! (k-1)
a183079_row n = a183079_tabf !! n
a183079_tabf = [1] : iterate (\row -> concatMap f row) [2]
   where f x = [a000217 x, a014132 x]
a183079_list = concat a183079_tabf
-- Reinhard Zumkeller, Dec 12 2012

tr[n_]:=n*(n+1)/2; nt[n_]:= n+Round@ Sqrt[2*n];a[1]=1; a[n_Integer] := a[n] = If[ EvenQ@n, nt@a[n/2], tr@ a@ Ceiling[n/2]]; a/@Range[58] (* Giovanni Resta, May 20 2015 *)

;; With memoizing definec-macro.
(definec (A183079 n) (cond ((<= n 1) n) ((even? n) (A014132 (A183079 (/ n 2)))) (else (A000217 (A183079 (/ (+ n 1) 2))))))
;; Antti Karttunen, May 18 2015

Let L(n) be the n-th triangular number (A000217).
Let U(n) be the n-th non-triangular number (A014132).
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; after which: a(2n) = A014132(a(n)), a(2n+1) = A000217(a(n+1)). - Antti Karttunen, May 20 2015

Formula added to the name and a new tree illustration to the Example section by Antti Karttunen, May 20 2015

A220348 Index of row where n occurs in A183079.

Original entry on oeis.org

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

A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(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, 9, 32, 48, 40, 28, 18, 13, 64, 96, 80, 56, 36, 26, 11, 128, 192, 160, 112, 72, 52, 22, 17, 256, 384, 320, 224, 144, 104, 44, 34, 25, 512, 768, 640, 448, 288, 208, 88, 68, 50, 21, 1024, 1536, 1280, 896, 576, 416, 176, 136, 100, 42, 15

Offset: 1

Antti Karttunen, May 18 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10440
Index entries for sequences that are permutations of the natural numbers

Inverse: A257798.
Cf. A000217, A002024, A010054, A083920.
Cf. also a similar permutation A220347 from which this differs for the first time at n=15, where a(15) = 9, while A220347(15) = 11.
a(n) differs from A054582(n-1) for the first time at n=21, where a(21) = 13, while A054582(20) = 11.

a(1) = 1; for n > 1: if A010054(n) = 1 [when n is triangular], then a(n) = 1+(2*a(A002024(n)-1)), otherwise a(n) = 2*a(A083920(n)).

Showing 1-3 of 3 results.

cp's OEIS Frontend

A183079 Tree generated by the triangular numbers: a(1) = 1; a(2n) = nontriangular(a(n)), a(2n+1) = triangular(a(n+1)), where triangular = A000217, nontriangular = A014132.

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A220348 Index of row where n occurs in A183079.

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A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(2a(n-1)), a(nontriangular(n)) = 2n, where triangular = A000217, nontriangular = A014132.

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cp's OEIS Frontend

A183079 Tree generated by the triangular numbers: a(1) = 1; a(2n) = nontriangular(a(n)), a(2n+1) = triangular(a(n+1)), where triangular = A000217, nontriangular = A014132.

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Examples

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Programs

Haskell

Mathematica

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Formula

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A220348 Index of row where n occurs in A183079.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Scheme

Scheme

Formula

Extensions

A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(2*a(n-1)), a(nontriangular(n)) = 2*n, where triangular = A000217, nontriangular = A014132.

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Formula

A257797 Permutation of natural numbers: a(1) = 1, a(triangular(n)) = 1+(2a(n-1)), a(nontriangular(n)) = 2n, where triangular = A000217, nontriangular = A014132.