login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A330375 Irregular triangle read by rows: T(n,k) (n>=1) is the sum of the lengths of all k-th right angles in all partitions of n. 3
1, 4, 9, 19, 1, 33, 2, 59, 7, 93, 12, 150, 26, 226, 43, 1, 342, 76, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column k starts in row k^2.

It appears that column 1 gives A179862.

LINKS

Table of n, a(n) for n=1..19.

EXAMPLE

Triangle begins:

    1;

    4;

    9;

   19,  1;

   33,  2;

   59,  7;

   93, 12;

  150, 26;

  226, 43, 1;

  342, 76, 2;

...

Figure 1 shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. Figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.

.

.                                     Right-angles   Right

Part   Ferrers diagram         Part   diagram        angle

                                      _ _ _ _ _ _ _

  7    * * * * * * *             7   |  _ _ _ _ _ _|  14

  6    * * * * * *               6   | |  _ _ _ _|     8

  3    * * *                     3   | | | |           2

  3    * * *                     3   | | |_|

  2    * *                       2   | |_|

  1    *                         1   | |

  1    *                         1   | |

  1    *                         1   |_|

.

       Figure 1.                      Figure 2.

.

For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:

.

    _       _ _       _ _ _       _ _ _ _       _ _ _ _ _

  1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8

  1| |    1| |      1| |        1| |          1| |

  1| |    1| |      1| |        1| |          1| |

  1| |    1| |      1| |        1| |          1|_|

  1| |    1| |      1| |        1|_|

  1| |    1| |      1|_|

  1| |    1|_|

  1|_|

    _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _

  6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8

  1| |              1|_|

  1|_|

.

    _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _

  2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7

  2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1

  1| |      1| |        1| |          1|_|

  1| |      1| |        1|_|

  1| |      1|_|

  1|_|

.

    _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _

  2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6

  2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2

  2| |_|    2| |_|      1| |        2|_|_|        1|_|

  1| |      1|_|        1|_|

  1|_|

.

    _ _       _ _ _        _ _ _ _

  2|  _|5   3|  _ _|5    4|  _ _ _|5

  2| | |3   3| |  _|3    4|_|_ _ _|3

  2| | |    2|_|_|

  2|_|_|

.

The sum of the lengths of the first right angles of all partitions of 8 is 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 7 + 7 + 7 + 7 + 7 + 6 + 6 + 6 + 6 + 6 + 6 + 5 + 5 + 5 = 150, so T(8,1) = 150.

The sum of the second right angles of all partitions of 8 is 1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 = 26, so T(8,2) = 26.

CROSSREFS

Row sums give A066186.

Cf. A179862.

Cf. also A000041, A330369, A330378, A330379.

Sequence in context: A146303 A344999 A203205 * A147977 A045278 A184723

Adjacent sequences:  A330372 A330373 A330374 * A330376 A330377 A330378

KEYWORD

nonn,tabf,more

AUTHOR

Omar E. Pol, Dec 21 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 29 23:00 EDT 2022. Contains 354913 sequences. (Running on oeis4.)