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

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, 25, 29, 55, 8, 12, 10, 18, 12, 16, 18, 34, 8, 12, 10, 18, 12, 16, 18, 34, 18, 26, 24, 44, 22, 30, 32, 60, 30, 46, 36, 64, 50, 66, 76, 144, 13, 19, 17, 31, 17, 23

Offset

0,4

Comments

The rule for constructing the tree is the following:

.....x

.....|

.....y

..../ \

..y+x..3y-x

and the tree begins like this:

.........0......

.........|......

.........1......

......./ \....

......1.....3....

...../ \.../ \...

....2...2.4...8..

and so on.

Column 1 : 0, 1, 1, 2, 3, 5, 8, ... = A000045 (Fibonacci numbers).

Column 2 : 3, 2, 5, 7, 12, 19, 31, ... = A013655.

Column 3 : 4, 3, 7, 10, 17, 27, 44, ... = A022120.

Column 4 : 8, 5, 13, 18, 31, 49, 80, ... = A022138.

Column 5 : 7, 5, 12, 17, 29, 46, 75, ... = A022137.

Column 6 : 9, 7, 16, 23, 39, 62, 101, ... = A190995.

Column 7 : 11, 7, 18, 25, 43, 68, 111, ... = A206419.

Column 8 : 21, 13, 34, 47, 81, 128, 209, ... = ?

Column 9 : 11, 8, 19, 27, 46, 73, 119, ... = A206420.

The lengths of the rows are 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ... = A011782 .

The final numbers in the rows are 0, 1, 3, 8, 21, 55, 144, ... = A001906.

The middle numbers in the rows are 1, 2, 5, 13, 34, 89, ... = A001519 .

Row sums for n>=1: 1, 4, 16, 64, 256, 1024, ... = 4^(n-1).

Links

Reinhard Zumkeller, Rows n = 0..13 of triangle, flattened

Example

The successive rows are:
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, 25, 29, 55
...

Maple

T:= proc(n, k) T(n, k):= `if`(k=1 and n<2, n, (d->(1+2*d)*
      T(n-1, r)+(1-2*d)*T(n-2, iquo(r+1, 2)))(irem(k+1, 2, 'r')))
    end:
seq(seq(T(n, k), k=1..max(1, 2^(n-1))), n=0..7); # Alois P. Heinz, Nov 07 2013

Mathematica

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]]];
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 *)

Prog

(Haskell)
data Dtree = Dtree Dtree (Integer, Integer) Dtree
a230871 n k = a230871_tabf !! n !! k
a230871_row n = a230871_tabf !! n
a230871_tabf = [0] : map (map snd) (rows $ deleham (0, 1)) where
   rows (Dtree left (x, y) right) =
        [(x, y)] : zipWith (++) (rows left) (rows right)
   deleham (x, y) = Dtree
           (deleham (y, y + x)) (x, y) (deleham (y, 3 * y - x))
-- Reinhard Zumkeller, Nov 07 2013

Crossrefs

Cf. A230872, A230873.

Cf. A231330, A231331, A231335.

Sequence in context: A325596 A254876 A249159 * A111241 A345055 A247501

Adjacent sequences: A230868 A230869 A230870 * A230872 A230873 A230874

Keyword

nonn,tabf

Author

Philippe Deléham, Nov 06 2013

Extensions

Incorrect fromula removed by Michel Marcus, Sep 23 2023

Status

approved