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

Author

Clark Kimberling, Dec 23 2010

Keywords

• nonn
• tabf

Comments

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 k

The 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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000217, A014132, A074049.

Cf. A220347 (inverse), A220348.

Cf. A183089, A183209 (similar permutations), also A257798.

Programs

Haskell

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

Mathematica

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 *)

Scheme

;; 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

Formula

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

Extensions

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

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