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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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, 13, ... = 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

FORMULA

T(n,k) = T(n-1,k) + T(n-2,k) for k < 2^(n-2), n > 1. - Philippe Deléham, Nov 07 2013

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 A247501 A192183

Adjacent sequences:  A230868 A230869 A230870 * A230872 A230873 A230874

KEYWORD

nonn,tabf

AUTHOR

Philippe Deléham, Nov 06 2013

STATUS

approved

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Last modified June 13 17:44 EDT 2021. Contains 345008 sequences. (Running on oeis4.)