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A253580 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).

%I A253580 #25 Dec 23 2024 14:53:44
%S A253580 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 A253580 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 A253580 15,13,11,9,7,5,3,1,0,2,4,6,8,10,12,14,16
%N A253580 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 A253580 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 A253580 T(n,n-k) + T(n,n+k) = 4*k - 1 for k = 1..n;
%C A253580 T(n+m,k) = T(n,k) for m > 0, k = 0 .. 2*n.
%D A253580 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 A253580 Reinhard Zumkeller, <a href="/A253580/b253580.txt">Rows n = 0..100 of triangle, flattened</a>
%H A253580 Éric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-January/014247.html">More fractal trees - and erasures</a>, SeqFan list, Jan 04 2015.
%e A253580 .   0:                                 0
%e A253580 .   1:                               1 0 2
%e A253580 .   2:                             3 1 0 2 4
%e A253580 .   3:                           5 3 1 0 2 4 6
%e A253580 .   4:                         7 5 3 1 0 2 4 6 8
%e A253580 .   5:                       9 7 5 3 1 0 2 4 6 8 10
%e A253580 .   6:                    11 9 7 5 3 1 0 2 4 6 8 10 12
%e A253580 .   7:                 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14
%e A253580 .   8:              15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14 16
%e A253580 .   9:           17 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14 16 18
%e A253580 .  10:        19 17 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14 16 18 20
%e A253580 .  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 A253580 .  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 A253580 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 A253580 (Haskell)
%o A253580 a253580 n k = a253580_tabf !! n !! k
%o A253580 a253580_row n = a253580_tabf !! n
%o A253580 a253580_tabf = [0] : [1,0,2] : f [1,0,2] where
%o A253580    f xs@(x:_) = ys : f ys where ys = [x + 2] ++ xs ++ [x + 3]
%o A253580 a253580_list = concat a253580_tabf
%o A253580 (PARI) a(n,k)=abs(n-k)*2-(k<n) \\ _M. F. Hasler_, Jan 04 2015
%Y A253580 Cf. A014105 (row sums), A253607 (first differences as flattened list), A253146.
%K A253580 nonn,tabf,easy,nice,look
%O A253580 0,4
%A A253580 _Eric Angelini_ and _Reinhard Zumkeller_, Jan 04 2015
%E A253580 Typo in definition corrected by _M. F. Hasler_, Jan 04 2015