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%I #27 Dec 31 2020 11:11:15
%S 1,2,2,3,3,4,4,5,3,5,6,6,7,7,8,8,8,9,9,10,10,11,5,5,11,12,12,13,5,13,
%T 14,6,6,14,15,15,16,16,17,7,7,17,18,12,18,19,19,20,8,8,20,21,21,22,22,
%U 23,32,23,24,24,25,7,25,26,10,10,26,27,27,28,8,8,28,29,11,11,29,30,30,31,31,32,12,26,12,32,33,9,9,33,34,34
%N Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(2n-1).
%C Row n lists the parts of the symmetric representation of A008438(n-1).
%C Also these are the parts from the odd-indexed rows of A237270.
%C Also these are the parts in the quadrants 1 and 3 of the spiral described in A239660, see example.
%C Row sums give A008438.
%C The length of row n is A237271(2n-1).
%C Both column 1 and the right border are equal to n.
%C Note that also the sequence can be represented in a quadrant.
%C We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016
%e 1;
%e 2, 2;
%e 3, 3;
%e 4, 4;
%e 5, 3, 5;
%e 6, 6;
%e 7, 7;
%e 8, 8, 8;
%e 9, 9;
%e 10, 10;
%e 11, 5, 5, 11;
%e 12, 12;
%e 13, 5, 13;
%e 14, 6, 6, 14;
%e 15, 15;
%e 16, 16;
%e 17, 7, 7, 17;
%e 18, 12, 18;
%e 19, 19;
%e 20, 8, 8, 20;
%e 21, 21;
%e 22, 22;
%e 23, 32, 23;
%e 24, 24;
%e 25, 7, 25;
%e ...
%e Illustration of initial terms (rows 1..8):
%e .
%e . _ _ _ _ _ _ _ 7
%e . |_ _ _ _ _ _ _|
%e . |
%e . |_ _
%e . _ _ _ _ _ 5 |_
%e . |_ _ _ _ _| |
%e . |_ _ 3 |_ _ _ 7
%e . |_ | | |
%e . _ _ _ 3 |_|_ _ 5 | |
%e . |_ _ _| | | | |
%e . |_ _ 3 | | | |
%e . | | | | | |
%e . _ 1 | | | | | |
%e . _ _ _ _ |_| |_| |_| |_|
%e . | | | | | | | |
%e . | | | | | | |_|_ _
%e . | | | | | | 2 |_ _|
%e . | | | | |_|_ 2
%e . | | | | 4 |_
%e . | | |_|_ _ |_ _ _ _
%e . | | 6 |_ |_ _ _ _|
%e . |_|_ _ _ |_ 4
%e . 8 | |_ _ |
%e . |_ | |_ _ _ _ _ _
%e . |_ |_ |_ _ _ _ _ _|
%e . 8 |_ _| 6
%e . |
%e . |_ _ _ _ _ _ _ _
%e . |_ _ _ _ _ _ _ _|
%e . 8
%e .
%e The figure shows the quadrants 1 and 3 of the spiral described in A239660.
%e For n = 5 we have that 2*5 - 1 = 9 and the 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5], so row 5 is [5, 3, 5], see the third arm of the spiral in the first quadrant.
%e The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the sum of the parts of the symmetric representation of sigma(9) is 5 + 3 + 5 = 13, equaling the sum of divisors of 9.
%Y Cf. A000203, A005408, A008438, A112610, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A239053, A239660, A239931, A239933, A244050, A245092, A262626.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Mar 31 2014
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