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A fractal tree, read by rows: for n > 1: T(n,0) = T(n-1,0)+2, T(n,2*n) = T(n-1,0)+3, and for k=1..2*n-1: T(n,k) = T(n-1,k-1).
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%I #24 Jan 02 2023 12:30:51

%S 0,1,0,2,3,1,0,2,4,5,3,1,0,2,4,6,7,5,3,1,0,2,4,6,8,9,7,5,3,1,0,2,4,6,

%T 8,10,11,9,7,5,3,1,0,2,4,6,8,10,12,13,11,9,7,5,3,1,0,2,4,6,8,10,12,14,

%U 15,13,11,9,7,5,3,1,0,2,4,6,8,10,12,14,16

%N A fractal tree, read by rows: for n > 1: T(n,0) = T(n-1,0)+2, T(n,2*n) = T(n-1,0)+3, and for k=1..2*n-1: T(n,k) = T(n-1,k-1).

%C If all pairs of consecutive terms (i,i+1) (such as (0,1), (1,2), (2,3), ...) are erased, the original sequence appears; see also A253607.

%C T(n,n-k) + T(n,n+k) = 4*k - 1 for k = 1..n;

%C T(n+m,k) = T(n,k) for m > 0, k = 0 .. 2*n.

%D V. A. Sankar Ponnapalli and V. Y. Jayasree Pappu, Design of Octagonal Fractal Array Antenna for Side Lobe Reduction with Morse-Thue Fractal Density Tapering Technique, Preprint, 2016.

%H Reinhard Zumkeller, <a href="/A253580/b253580.txt">Rows n = 0..100 of triangle, flattened</a>

%H Éric Angelini, <a href="http://list.seqfan.eu/oldermail/seqfan/2015-January/014247.html">More fractal trees - and erasures</a>, SeqFan list, Jan 04 2015.

%e . 0: 0

%e . 1: 1 0 2

%e . 2: 3 1 0 2 4

%e . 3: 5 3 1 0 2 4 6

%e . 4: 7 5 3 1 0 2 4 6 8

%e . 5: 9 7 5 3 1 0 2 4 6 8 10

%e . 6: 11 9 7 5 3 1 0 2 4 6 8 10 12

%e . 7: 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14

%e . 8: 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14 16

%e . 9: 17 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14 16 18

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

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

%e . 12: 23 21 19 17 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14 16 18 20 22 24 .

%t a[n_, k_] := 2 Abs[n-k] - Boole[k<n]; Table[a[n, k], {n, 0, 9}, {k, 0, 2 n}] // Flatten (* _Jean-François Alcover_, Nov 04 2016, after _M. F. Hasler_ *)

%o (Haskell)

%o a253580 n k = a253580_tabf !! n !! k

%o a253580_row n = a253580_tabf !! n

%o a253580_tabf = [0] : [1,0,2] : f [1,0,2] where

%o f xs@(x:_) = ys : f ys where ys = [x + 2] ++ xs ++ [x + 3]

%o a253580_list = concat a253580_tabf

%o (PARI) a(n,k)=abs(n-k)*2-(k<n) \\ _M. F. Hasler_, Jan 04 2015

%Y Cf. A014105 (row sums), A253607 (first differences as flattened list), A253146.

%K nonn,tabf,easy,nice,look

%O 0,4

%A _Eric Angelini_ and _Reinhard Zumkeller_, Jan 04 2015

%E Typo in definition corrected by _M. F. Hasler_, Jan 04 2015