login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

a(n) is the sum over all partitions of n of the number of right angles that are not the largest right angle.
3

%I #30 Apr 11 2024 04:51:28

%S 0,0,0,1,2,5,8,14,22,34,50,75,106,151,210,291,394,535,712,949,1246,

%T 1634,2118,2745,3520,4508,5728,7266,9152,11512,14390,17959,22298,

%U 27634,34094,41993,51510,63075,76966,93752,113834,137992,166788,201269,242248,291102,348976,417727

%N a(n) is the sum over all partitions of n of the number of right angles that are not the largest right angle.

%C a(n) is also the sum of the sizes of the Durfee squares of all partitions of n, minus the number of partitions of n.

%C a(n) is also the sum of positive cranks of all partitions of n, minus the number of partitions of n.

%D G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143.

%F a(n) = A115995(n) - A000041(n), n >= 1.

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

%e .

%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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

%e 1| | 1| | 1| | 1| | 1| |

%e 1| | 1| | 1| | 1| | 1| |

%e 1| | 1| | 1| | 1| | 1|_|

%e 1| | 1| | 1| | 1|_|

%e 1| | 1| | 1|_|

%e 1| | 1|_|

%e 1|_|

%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

%e 6| _ _ _ _ _|8 7| _ _ _ _ _ _|8 8|_ _ _ _ _ _ _ _|8

%e 1| | 1|_|

%e 1|_|

%e .

%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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

%e 2| |_|1 2| |_| 1 2| |_| 1 2| |_| 1 2|_|_| 1

%e 1| | 1| | 1| | 1|_|

%e 1| | 1| | 1|_|

%e 1| | 1|_|

%e 1|_|

%e .

%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

%e 2| _|6 3| _ _|6 3| _ _|6 4| _ _ _|6 4| _ _ _|6 5| _ _ _ _|6

%e 2| | |2 2| | | 2 3| |_ _|2 2| | | 2 3| |_ _| 2 3|_|_ _| 2

%e 2| |_| 2| |_| 1| | 2|_|_| 1|_|

%e 1| | 1|_| 1|_|

%e 1|_|

%e .

%e _ _ _ _ _ _ _ _ _

%e 2| _|5 3| _ _|5 4| _ _ _|5

%e 2| | |3 3| | _|3 4|_|_ _ _|3

%e 2| | | 2|_|_|

%e 2|_|_|

%e .

%e In total there are 14 right angles that are not the largest right angle of the partitions of 8, so a(8) = 14.

%Y Cf. A000041, A115995, A330369, A330375, A330379.

%K nonn

%O 1,5

%A _Omar E. Pol_, Jan 01 2020

%E a(17) and a(47) corrected by _Georg Fischer_, Apr 11 2024