cp's OEIS Frontend

A239934 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n).

%I A239934 #48 Sep 21 2018 08:14:10
%S A239934 7,15,28,31,42,60,56,63,91,90,42,42,124,49,49,120,168,127,63,63,195,
%T A239934 70,70,186,224,180,84,84,252,217,210,280,248,105,105,360,112,112,255
%N A239934 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n).
%C A239934 Row n is a palindromic composition of sigma(4n).
%C A239934 Row n is also the row 4n of A237270.
%C A239934 Row n has length A237271(4n).
%C A239934 Row sums give A193553.
%C A239934 First differs from A193553 at a(11).
%C A239934 Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the fourth quadrant of the spiral described in A239660, see example.
%C A239934 For the parts of the symmetric representation of sigma(4n-3), see A239931.
%C A239934 For the parts of the symmetric representation of sigma(4n-2), see A239932.
%C A239934 For the parts of the symmetric representation of sigma(4n-1), see A239933.
%C A239934 We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - _Omar E. Pol_, Dec 06 2016
%e A239934 The irregular triangle begins:
%e A239934     7;
%e A239934    15;
%e A239934    28;
%e A239934    31;
%e A239934    42;
%e A239934    60;
%e A239934    56;
%e A239934    63;
%e A239934    91;
%e A239934    90;
%e A239934    42, 42;
%e A239934   124;
%e A239934    49, 49;
%e A239934   120;
%e A239934   168;
%e A239934   ...
%e A239934 Illustration of initial terms in the fourth quadrant of the spiral described in A239660:
%e A239934 .
%e A239934 .           7       15      28      31      42      60      56      63
%e A239934 .           _       _       _       _       _       _       _       _
%e A239934 .          | |     | |     | |     | |     | |     | |     | |     | |
%e A239934 .         _| |     | |     | |     | |     | |     | |     | |     | |
%e A239934 .     _ _|  _|     | |     | |     | |     | |     | |     | |     | |
%e A239934 .    |_ _ _|    _ _| |     | |     | |     | |     | |     | |     | |
%e A239934 .             _|  _ _|     | |     | |     | |     | |     | |     | |
%e A239934 .            |  _|    _ _ _| |     | |     | |     | |     | |     | |
%e A239934 .     _ _ _ _| |    _|    _ _|     | |     | |     | |     | |     | |
%e A239934 .    |_ _ _ _ _|  _|     |    _ _ _| |     | |     | |     | |     | |
%e A239934 .                |      _|   |  _ _ _|     | |     | |     | |     | |
%e A239934 .                |  _ _|    _| |    _ _ _ _| |     | |     | |     | |
%e A239934 .     _ _ _ _ _ _| |      _|  _|   |  _ _ _ _|     | |     | |     | |
%e A239934 .    |_ _ _ _ _ _ _|  _ _|  _|  _ _| |    _ _ _ _ _| |     | |     | |
%e A239934 .                    |  _ _|  _|    _|   |    _ _ _ _|     | |     | |
%e A239934 .                    | |     |     |  _ _|   |    _ _ _ _ _| |     | |
%e A239934 .     _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |   |  _ _ _ _ _|     | |
%e A239934 .    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|   | |    _ _ _ _ _ _| |
%e A239934 .                        | |     |      _|    _ _| |   |  _ _ _ _ _ _|
%e A239934 .                        | |  _ _|    _|  _ _|  _ _|   | |
%e A239934 .     _ _ _ _ _ _ _ _ _ _| | |       |   |    _|    _ _| |
%e A239934 .    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|  _|     |  _ _|
%e A239934 .                            | |       |  _|      _| |
%e A239934 .                            | |  _ _ _| |      _|  _|
%e A239934 .     _ _ _ _ _ _ _ _ _ _ _ _| | |  _ _ _|  _ _|  _|
%e A239934 .    |_ _ _ _ _ _ _ _ _ _ _ _ _| | |       |  _ _|
%e A239934 .                                | |  _ _ _| |
%e A239934 .                                | | |  _ _ _|
%e A239934 .     _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
%e A239934 .    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e A239934 .                                    | |
%e A239934 .                                    | |
%e A239934 .     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A239934 .    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A239934 .
%e A239934 For n = 7 we have that 4*7 = 28 and the 28th row of A237593 is [15, 5, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 5, 15] and the 27th row of A237593 is [14, 5, 3, 2, 1, 2, 2, 1, 2, 3, 5, 14] therefore between both Dyck paths there are only one region (or part) of size 56, so row 7 is 56.
%e A239934 The sum of divisors of 28 is 1 + 2 + 4 + 7 + 14 + 28 = A000203(28) = 56. On the other hand the sum of the parts of the symmetric representation of sigma(28) is 56, equaling the sum of divisors of 28.
%e A239934 For n = 11 we have that 4*11 = 44 and the 44th row of A237593 is [23, 8, 4, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 8, 23] and the 43rd row of A237593 is [22, 8, 4, 3, 2, 1, 2, 1, 1, 2, 1, 2, 3, 4, 8, 23] therefore between both Dyck paths there are two regions (or parts) of sizes [42, 42], so row 11 is [42, 42].
%e A239934 The sum of divisors of 44 is 1 + 2 + 4 + 11 + 22 + 44 = A000203(44) = 84. On the other hand the sum of the parts of the symmetric representation of sigma(44) is 42 + 42 = 84, equaling the sum of divisors of 44.
%Y A239934 Cf. A000203, A193553, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A239660, A239931, A239932, A239933, A244050, A245092, A262626.
%K A239934 nonn,tabf,more
%O A239934 1,1
%A A239934 _Omar E. Pol_, Mar 29 2014