1,1

Row n is a palindromic composition of sigma(4n).

Row n is also the row 4n of A237270.

Row n has length A237271(4n).

Row sums give A193553.

First differs from A193553 at a(11).

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.

For the parts of the symmetric representation of sigma(4n-3), see A239931.

For the parts of the symmetric representation of sigma(4n-2), see A239932.

For the parts of the symmetric representation of sigma(4n-1), see A239933.

We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

Table of n, a(n) for n=1..39.

The irregular triangle begins:

    7;

   15;

   28;

   31;

   42;

   60;

   56;

   63;

   91;

   90;

   42, 42;

  124;

   49, 49;

  120;

  168;

  ...

Illustration of initial terms in the fourth quadrant of the spiral described in A239660:

.

.           7       15      28      31      42      60      56      63

.           _       _       _       _       _       _       _       _

.          | |     | |     | |     | |     | |     | |     | |     | |

.         _| |     | |     | |     | |     | |     | |     | |     | |

.     _ _|  _|     | |     | |     | |     | |     | |     | |     | |

.    |_ _ _|    _ _| |     | |     | |     | |     | |     | |     | |

.             _|  _ _|     | |     | |     | |     | |     | |     | |

.            |  _|    _ _ _| |     | |     | |     | |     | |     | |

.     _ _ _ _| |    _|    _ _|     | |     | |     | |     | |     | |

.    |_ _ _ _ _|  _|     |    _ _ _| |     | |     | |     | |     | |

.                |      _|   |  _ _ _|     | |     | |     | |     | |

.                |  _ _|    _| |    _ _ _ _| |     | |     | |     | |

.     _ _ _ _ _ _| |      _|  _|   |  _ _ _ _|     | |     | |     | |

.    |_ _ _ _ _ _ _|  _ _|  _|  _ _| |    _ _ _ _ _| |     | |     | |

.                    |  _ _|  _|    _|   |    _ _ _ _|     | |     | |

.                    | |     |     |  _ _|   |    _ _ _ _ _| |     | |

.     _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |   |  _ _ _ _ _|     | |

.    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|   | |    _ _ _ _ _ _| |

.                        | |     |      _|    _ _| |   |  _ _ _ _ _ _|

.                        | |  _ _|    _|  _ _|  _ _|   | |

.     _ _ _ _ _ _ _ _ _ _| | |       |   |    _|    _ _| |

.    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|  _|     |  _ _|

.                            | |       |  _|      _| |

.                            | |  _ _ _| |      _|  _|

.     _ _ _ _ _ _ _ _ _ _ _ _| | |  _ _ _|  _ _|  _|

.    |_ _ _ _ _ _ _ _ _ _ _ _ _| | |       |  _ _|

.                                | |  _ _ _| |

.                                | | |  _ _ _|

.     _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |

.    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |

.                                    | |

.                                    | |

.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |

.    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|

.

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.

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.

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].

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.

Cf. A000203, A193553, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A239660, A239931, A239932, A239933, A244050, A245092, A262626.

Sequence in context: A063611 A292379 A146624 * A193553 A274090 A195041

Adjacent sequences:  A239931 A239932 A239933 * A239935 A239936 A239937

nonn,tabf,more

Omar E. Pol, Mar 29 2014

approved