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A253672 Another fractal t(h)ree.

%I A253672 #20 Dec 23 2024 14:53:44
%S A253672 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,
%T A253672 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,
%U A253672 5,2,0,1,3,4,6,7,9,10,12,13,15,16,18,19,20
%N A253672 Another fractal t(h)ree.
%C A253672 A008585(n+1) = length of row n; A062741(n+1) = sum of row n;
%C A253672 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.
%H A253672 Reinhard Zumkeller, <a href="/A253672/b253672.txt">Rows n = 0..100 of triangle, flattened</a>
%H A253672 Éric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-January/014268.html">More (and more) fractal trees - and erasures</a>, SeqFan list, Jan 08 2015.
%F A253672 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.
%e A253672 .  0:                                     | 0  1  2|
%e A253672 .  1:                                0  1 | 3  4  5|  2
%e A253672 .  2:                          0  1  3  4 | 6  7  8|  5  2
%e A253672 .  3:                    0  1  3  4  6  7 | 9 10 11|  8  5  2
%e A253672 .  4:              0  1  3  4  6  7  9 10 |12 13 14| 11  8  5  2
%e A253672 .  5:          0 1 3  4  6  7  9 10 12 13 |15 16 17| 14 11  8  5 2
%e A253672 .  6:      0 1 3 4 6  7  9 10 12 13 15 16 |18 19 20| 17 14 11  8 5 2
%e A253672 .  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 .
%o A253672 (Haskell)
%o A253672 a253672 n k = a253672_tabf !! n !! k
%o A253672 a253672_row n = a253672_tabf !! n
%o A253672 a253672_tabf = [0,1,2] : f [] [0,1,2] [] (iterate (map (+ 3)) [3..5]) where
%o A253672    f as bs cs (uvws:uvwss) = (as' ++ uvws ++ cs') : f as' uvws cs' uvwss
%o A253672      where as' = as ++ [u,v]; cs' = [w] ++ cs
%o A253672            [u,v,w] = bs
%o A253672 a253672_list = concat a253672_tabf
%Y A253672 Cf. A008585, A062741.
%K A253672 nonn,tabf
%O A253672 0,3
%A A253672 _Eric Angelini_ and _Reinhard Zumkeller_, Jan 08 2015