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Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.
6

%I #58 Nov 11 2024 10:54:23

%S 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 25,29,55,8,12,10,18,12,16,18,34,8,12,10,18,12,16,18,34,18,26,24,44,

%U 22,30,32,60,30,46,36,64,50,66,76,144,13,19,17,31,17,23

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

%C The rule for constructing the tree is the following:

%C .....x

%C .....|

%C .....y

%C ..../ \

%C ..y+x..3y-x

%C and the tree begins like this:

%C .........0......

%C .........|......

%C .........1......

%C ......./ \....

%C ......1.....3....

%C ...../ \.../ \...

%C ....2...2.4...8..

%C and so on.

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

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

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

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

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

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

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

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

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

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

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

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

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

%H Reinhard Zumkeller, <a href="/A230871/b230871.txt">Rows n = 0..13 of triangle, flattened</a>

%e The successive rows are:

%e 0

%e 1

%e 1, 3

%e 2, 2, 4, 8

%e 3, 5, 3, 5, 7, 9, 11, 21

%e 5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55

%e ...

%p T:= proc(n, k) T(n, k):= `if`(k=1 and n<2, n, (d->(1+2*d)*

%p T(n-1, r)+(1-2*d)*T(n-2, iquo(r+1, 2)))(irem(k+1, 2, 'r')))

%p end:

%p seq(seq(T(n, k), k=1..max(1, 2^(n-1))), n=0..7); # _Alois P. Heinz_, Nov 07 2013

%t 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 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 (Haskell)

%o data Dtree = Dtree Dtree (Integer, Integer) Dtree

%o a230871 n k = a230871_tabf !! n !! k

%o a230871_row n = a230871_tabf !! n

%o a230871_tabf = [0] : map (map snd) (rows $ deleham (0, 1)) where

%o rows (Dtree left (x, y) right) =

%o [(x, y)] : zipWith (++) (rows left) (rows right)

%o deleham (x, y) = Dtree

%o (deleham (y, y + x)) (x, y) (deleham (y, 3 * y - x))

%o -- _Reinhard Zumkeller_, Nov 07 2013

%Y Cf. A230872, A230873.

%Y Cf. A231330, A231331, A231335.

%K nonn,tabf

%O 0,4

%A _Philippe Deléham_, Nov 06 2013

%E Incorrect formula removed by _Michel Marcus_, Sep 23 2023