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%I #87 Apr 23 2023 07:24:00
%S 1,1,1,1,2,2,2,2,2,1,1,2,2,1,1,2,3,1,1,3,3,1,1,3,3,2,2,3,3,2,2,3,4,1,
%T 1,1,1,4,4,1,1,1,1,4,4,2,1,1,2,4,4,2,1,1,2,4,5,2,1,1,2,5,5,2,1,1,2,5,
%U 5,2,2,2,2,5,5,2,2,2,2,5,6,2,1,1,1,1,2,6,6,2,1,1,1,1,2,6,6,3,1,1,1,1,3,6,6,3,1,1,1,1,3,6
%N Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593.
%C For the construction of this sequence also we can start from A235791.
%C This sequence can be interpreted as an infinite Dyck path: UDUDUUDD...
%C Also we use this sequence for the construction of a spiral in which the arms in the quadrants give the symmetric representation of sigma, see example.
%C We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - _Omar E. Pol_, Dec 07 2016
%C The spiral has the property that the sum of the parts in the quadrants 1 and 3, divided by the sum of the parts in the quadrants 2 and 4, converges to 3/5. - _Omar E. Pol_, Jun 10 2019
%H Robert Price, <a href="/A239660/b239660.txt">Table of n, a(n) for n = 1..30016</a> (rows n = 1..412, flattened)
%e Triangle begins (first 15.5 rows):
%e 1, 1, 1, 1;
%e 2, 2, 2, 2;
%e 2, 1, 1, 2, 2, 1, 1, 2;
%e 3, 1, 1, 3, 3, 1, 1, 3;
%e 3, 2, 2, 3, 3, 2, 2, 3;
%e 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4;
%e 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4;
%e 5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5;
%e 5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5;
%e 6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6;
%e 6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6;
%e 7, 2, 2, 1, 1, 2, 2, 7, 7, 2, 2, 1, 1, 2, 2, 7;
%e 7, 3, 2, 1, 1, 2, 3, 7, 7, 3, 2, 1, 1, 2, 3, 7;
%e 8, 3, 1, 2, 2, 1, 3, 8, 8, 3, 1, 2, 2, 1, 3, 8;
%e 8, 3, 2, 1, 1, 1, 1, 2, 3, 8, 8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
%e 9, 3, 2, 1, 1, 1, 1, 2, 3, 9, ...
%e .
%e Illustration of initial terms as an infinite Dyck path (row n = 1..4):
%e .
%e . /\/\ /\/\
%e . /\ /\ /\/\ /\/\ / \ / \
%e . /\/\/ \/ \/ \/ \/ \/ \
%e .
%e .
%e Illustration of initial terms for the construction of a spiral related to sigma:
%e .
%e . row 1 row 2 row 3 row 4
%e . _ _ _
%e . |_
%e . _ _ |
%e . _ _ | |
%e . | | | |
%e . | | | |
%e . |_ _ |_ _| |
%e . |_ _ _ _| _|
%e . _ _ _|
%e .
%e .[1,1,1,1] [2,2,2,2] [2,1,1,2,2,1,1,2] [3,1,1,3,3,1,1,3]
%e .
%e The first 2*A003056(n) terms of the n-th row are represented in the A010883(n-1) quadrant and the last 2*A003056(n) terms of the n-th row are represented in the A010883(n) quadrant.
%e .
%e Illustration of the spiral constructed with the first 15.5 rows of triangle:
%e .
%e . 12 _ _ _ _ _ _ _ _
%e . | _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
%e . | | |_ _ _ _ _ _ _|
%e . _| | |
%e . |_ _|9 _ _ _ _ _ _ |_ _
%e . 12 _ _| | _ _ _ _ _|_ _ _ _ _ 5 |_
%e . _ _ _| | _| | |_ _ _ _ _| |
%e . | _ _ _| 9 _|_ _| |_ _ 3 |_ _ _ 7
%e . | | _ _| | 12 _ _ _ _ |_ | | |
%e . | | | _ _| _| _ _ _|_ _ _ 3 |_|_ _ 5 | |
%e . | | | | _| | |_ _ _| | | | |
%e . | | | | | _ _| |_ _ 3 | | | |
%e . | | | | | | 3 _ _ | | | | | |
%e . | | | | | | | _|_ 1 | | | | | |
%e . _|_| _|_| _|_| _|_| |_| _|_| _|_| _|_| _
%e . | | | | | | | | | | | | | | | |
%e . | | | | | | |_|_ _ _| | | | | | | |
%e . | | | | | | 2 |_ _|_ _| _| | | | | | |
%e . | | | | |_|_ 2 |_ _ _| _ _| | | | | |
%e . | | | | 4 |_ 7 _| _ _| | | | |
%e . | | |_|_ _ |_ _ _ _ | _| _ _ _| | | |
%e . | | 6 |_ |_ _ _ _|_ _ _ _| | _| _ _| | |
%e . |_|_ _ _ |_ 4 |_ _ _ _ _| _| | _ _ _| |
%e . 8 | |_ _ | 15| _| | _ _ _|
%e . |_ | |_ _ _ _ _ _ | _ _| _| |
%e . 8 |_ |_ |_ _ _ _ _ _|_ _ _ _ _ _| | _| _|
%e . |_ _| 6 |_ _ _ _ _ _ _| _ _| _|
%e . | 28| _ _|
%e . |_ _ _ _ _ _ _ _ | |
%e . |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
%e . 8 |_ _ _ _ _ _ _ _ _|
%e . 31
%e .
%e The diagram contains A237590(16) = 27 parts.
%e The total area (also the total number of cells) in the n-th arm of the spiral is equal to sigma(n) = A000203(n), considering every quadrant and the axes x and y. (checked by hand up to row n = 128). The parts of the spiral are in A237270: 1, 3, 2, 2, 7...
%e Diagram extended by _Omar E. Pol_, Aug 23 2018
%Y Row n has length 4*A003056(n).
%Y The sum of row n is equal to 4*n = A008586(n).
%Y Row n is a palindromic composition of 4*n = A008586(n).
%Y Both column 1 and right border are A008619, n >= 1.
%Y The connection between A196020 and A237270 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> this sequence --> A237270.
%Y Cf. A000203, A000217, A003056, A008619, A010883, A112610, A193553, A237048, A237271, A237590, A239052, A239053, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A262626, A296508, A299778.
%K nonn,look,tabf
%O 1,5
%A _Omar E. Pol_, Mar 24 2014
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