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%I #30 Sep 09 2017 19:39:36
%S 1,2,3,4,6,5,10,7,21,9,15,8,55,14,28,11,231,27,45,13,120,20,36,12,
%T 1540,65,105,19,406,35,66,16,26796,252,378,34,1035,54,91,18,7260,135,
%U 210,26,666,44,78,17,1186570,1595,2145,76,5565,119,190,25,82621,434
%N 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.
%C A permutation of the positive integers.
%C In general, suppose that L and U are complementary sequences of positive integers such that
%C (1) L(1)=1; and
%C (2) if n>1, then n=L(k) or n=U(k) for some k<n.
%C The tree generated by the sequence L is defined as follows:
%C T(0,0)=1; T(1,0)=2; T(n,2j)=L(T(n-1,j));
%C T(n,2j+1)=U(T(n-1,j)); for j=0,1,...,2^(n-1)-1, n>=2.
%C The numbers, taken in the order generated, form a permutation of the positive integers.
%H Reinhard Zumkeller, <a href="/A183079/b183079.txt">Rows n = 1..14 of triangle, flattened</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F Let L(n) be the n-th triangular number (A000217).
%F Let U(n) be the n-th non-triangular number (A014132).
%F The tree-array T(n,k) is then given by rows:
%F T(0,0)=1; T(1,0)=2;
%F T(n,2j)=L(T(n-1,j));
%F T(n,2j+1)=U(T(n-1,j));
%F for j=0,1,...,2^(n-1)-1, n>=2.
%F a(1) = 1; after which: a(2n) = A014132(a(n)), a(2n+1) = A000217(a(n+1)). - _Antti Karttunen_, May 20 2015
%e First levels of the tree:
%e 1
%e |
%e ...................2...................
%e 3 4
%e 6......../ \........5 10......./ \........7
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e 21 9 15 8 55 14 28 11
%e 231 27 45 13 120 20 36 12 1540 65 105 19 406 35 66 16
%e Beginning with 3 and 4, the numbers are generated in pairs, such as (3,4), (6,5), (10,7), (21,9),...
%e In all such pairs, the first number belongs to A000217; the second, to A014132.
%t 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 *)
%o (Haskell)
%o a183079 n k = a183079_tabf !! (n-1) !! (k-1)
%o a183079_row n = a183079_tabf !! n
%o a183079_tabf = [1] : iterate (\row -> concatMap f row) [2]
%o where f x = [a000217 x, a014132 x]
%o a183079_list = concat a183079_tabf
%o -- _Reinhard Zumkeller_, Dec 12 2012
%o (Scheme, with memoizing definec-macro)
%o (definec (A183079 n) (cond ((<= n 1) n) ((even? n) (A014132 (A183079 (/ n 2)))) (else (A000217 (A183079 (/ (+ n 1) 2))))))
%o ;; _Antti Karttunen_, May 18 2015
%Y Cf. A000217, A014132, A074049.
%Y Cf. A220347 (inverse), A220348.
%Y Cf. A183089, A183209 (similar permutations), also A257798.
%K nonn,tabf
%O 1,2
%A _Clark Kimberling_, Dec 23 2010
%E Formula added to the name and a new tree illustration to the Example section by _Antti Karttunen_, May 20 2015
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