A253672 Another fractal t(h)ree.

0, 1, 2, 0, 1, 3, 4, 5, 2, 0, 1, 3, 4, 6, 7, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 11, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 11, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 17, 14, 11, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 20

Offset: 0

Author

Eric Angelini and Reinhard Zumkeller, Jan 08 2015

Keywords

• nonn
• tabf

Comments

A008585(n+1) = length of row n; A062741(n+1) = sum of row n;

the fractal nature is illustrated by the following manipulation: remove from all rows the first two terms and also the last one, after subtracting all terms by 3, the original t(h)ree will reappear.

Examples

			.  0:                                     | 0  1  2|
.  1:                                0  1 | 3  4  5|  2
.  2:                          0  1  3  4 | 6  7  8|  5  2
.  3:                    0  1  3  4  6  7 | 9 10 11|  8  5  2
.  4:              0  1  3  4  6  7  9 10 |12 13 14| 11  8  5  2
.  5:          0 1 3  4  6  7  9 10 12 13 |15 16 17| 14 11  8  5 2
.  6:      0 1 3 4 6  7  9 10 12 13 15 16 |18 19 20| 17 14 11  8 5 2
.  7:  0 1 3 4 6 7 9 10 12 13 15 16 18 19 |21 22 23| 20 17 14 11 8 5 2 .
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
• Éric Angelini, More (and more) fractal trees - and erasures, SeqFan list, Jan 08 2015.

Crossrefs

Cf. A008585, A062741.

Programs

Haskell

a253672 n k = a253672_tabf !! n !! k
a253672_row n = a253672_tabf !! n
a253672_tabf = [0,1,2] : f [] [0,1,2] [] (iterate (map (+ 3)) [3..5]) where
   f as bs cs (uvws:uvwss) = (as' ++ uvws ++ cs') : f as' uvws cs' uvwss
     where as' = as ++ [u,v]; cs' = [w] ++ cs
           [u,v,w] = bs
a253672_list = concat a253672_tabf

Formula

T(n,0) = 0; T(n,1)=1; T(n,2*n-1) = 2; T(n+1,k+2) = T(n,k)+3, k = 0..3*n-1.