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A230871 Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.

%I A230871 #58 Nov 11 2024 10:54:23
%S A230871 0,1,1,3,2,2,4,8,3,5,3,5,7,9,11,21,5,7,7,13,5,7,7,13,11,17,13,23,19,
%T A230871 25,29,55,8,12,10,18,12,16,18,34,8,12,10,18,12,16,18,34,18,26,24,44,
%U A230871 22,30,32,60,30,46,36,64,50,66,76,144,13,19,17,31,17,23
%N A230871 Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.
%C A230871 The rule for constructing the tree is the following:
%C A230871 .....x
%C A230871 .....|
%C A230871 .....y
%C A230871 ..../ \
%C A230871 ..y+x..3y-x
%C A230871 and the tree begins like this:
%C A230871 .........0......
%C A230871 .........|......
%C A230871 .........1......
%C A230871 ......./   \....
%C A230871 ......1.....3....
%C A230871 ...../ \.../ \...
%C A230871 ....2...2.4...8..
%C A230871 and so on.
%C A230871 Column 1 :  0,  1,  1,  2,  3,   5,   8, ... = A000045 (Fibonacci numbers).
%C A230871 Column 2 :  3,  2,  5,  7, 12,  19,  31, ... = A013655.
%C A230871 Column 3 :  4,  3,  7, 10, 17,  27,  44, ... = A022120.
%C A230871 Column 4 :  8,  5, 13, 18, 31,  49,  80, ... = A022138.
%C A230871 Column 5 :  7,  5, 12, 17, 29,  46,  75, ... = A022137.
%C A230871 Column 6 :  9,  7, 16, 23, 39,  62, 101, ... = A190995.
%C A230871 Column 7 : 11,  7, 18, 25, 43,  68, 111, ... = A206419.
%C A230871 Column 8 : 21, 13, 34, 47, 81, 128, 209, ... = ?
%C A230871 Column 9 : 11,  8, 19, 27, 46,  73, 119, ... = A206420.
%C A230871 The lengths of the rows are 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ... = A011782 .
%C A230871 The final numbers in the rows are 0, 1, 3, 8, 21, 55, 144, ... = A001906.
%C A230871 The middle numbers in the rows are 1, 2, 5, 13, 34, 89, ... =  A001519 .
%C A230871 Row sums for n>=1: 1, 4, 16, 64, 256, 1024, ... = 4^(n-1).
%H A230871 Reinhard Zumkeller, <a href="/A230871/b230871.txt">Rows n = 0..13 of triangle, flattened</a>
%e A230871 The successive rows are:
%e A230871   0
%e A230871   1
%e A230871   1, 3
%e A230871   2, 2, 4, 8
%e A230871   3, 5, 3, 5, 7, 9, 11, 21
%e A230871   5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55
%e A230871   ...
%p A230871 T:= proc(n, k) T(n, k):= `if`(k=1 and n<2, n, (d->(1+2*d)*
%p A230871       T(n-1, r)+(1-2*d)*T(n-2, iquo(r+1, 2)))(irem(k+1, 2, 'r')))
%p A230871     end:
%p A230871 seq(seq(T(n, k), k=1..max(1, 2^(n-1))), n=0..7); # _Alois P. Heinz_, Nov 07 2013
%t A230871 T[n_, k_] := T[n, k] = If[k==1 && n<2, n, Function[d, r = Quotient[k+1, 2]; (1+2d) T[n-1, r] + (1-2d) T[n-2, Quotient[r+1, 2]]][Mod[k+1, 2]]];
%t A230871 Table[T[n, k], {n, 0, 7}, {k, 1, Max[1, 2^(n-1)]}] // Flatten (* _Jean-François Alcover_, Apr 11 2017, after _Alois P. Heinz_ *)
%o A230871 (Haskell)
%o A230871 data Dtree = Dtree Dtree (Integer, Integer) Dtree
%o A230871 a230871 n k = a230871_tabf !! n !! k
%o A230871 a230871_row n = a230871_tabf !! n
%o A230871 a230871_tabf = [0] : map (map snd) (rows $ deleham (0, 1)) where
%o A230871    rows (Dtree left (x, y) right) =
%o A230871         [(x, y)] : zipWith (++) (rows left) (rows right)
%o A230871    deleham (x, y) = Dtree
%o A230871            (deleham (y, y + x)) (x, y) (deleham (y, 3 * y - x))
%o A230871 -- _Reinhard Zumkeller_, Nov 07 2013
%Y A230871 Cf. A230872, A230873.
%Y A230871 Cf. A231330, A231331, A231335.
%K A230871 nonn,tabf
%O A230871 0,4
%A A230871 _Philippe Deléham_, Nov 06 2013
%E A230871 Incorrect formula removed by _Michel Marcus_, Sep 23 2023