%I #39 Dec 05 2023 17:01:13

%S 4,4,8,12,20,28,44,60,88,120,168,224,308,404,540,704,924,1188,1540,

%T 1960,2508,3168,4008,5020,6300,7832,9744,12040,14872,18260,22416,

%U 27368,33396,40572,49240,59532,71908,86548,104060,124740,149352,178332,212696,253044,300700,356536,422232,499016,589092,694100,816904

%N a(n) = 4*p(n), where p(n) is the number of partitions of n.

%C For n >= 1, a(n) is also the number of edges in the diagram of partitions of n, in which A299475(n) is the number of vertices and A000041(n) is the number of regions (see example and Euler's formula).

%H Shawn A. Broyles, <a href="/A299474/b299474.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 4*A000041(n) = 2*A139582(n).

%F a(n) = A000041(n) + A299475(n) - 1, n >= 1 (Euler's formula).

%F a(n) = A000041(n) + A299473(n). - _Omar E. Pol_, Aug 11 2018

%e Construction of a modular table of partitions in which a(n) is the number of edges of the diagram after n-th stage (n = 1..6):

%e --------------------------------------------------------------------------------

%e n ........: 1 2 3 4 5 6 (stage)

%e a(n)......: 4 8 12 20 28 44 (edges)

%e A299475(n): 4 7 10 16 22 34 (vertices)

%e A000041(n): 1 2 3 5 7 11 (regions)

%e --------------------------------------------------------------------------------

%e r p(n)

%e --------------------------------------------------------------------------------

%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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

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

%e 3 .... 3 ................|_ _ _| |_ _ _| | |_ _ _| | | |_ _ _| | | |

%e 4 |_ _| | |_ _| | | |_ _| | | |

%e 5 .... 5 .........................|_ _ _ _| |_ _ _ _| | |_ _ _ _| | |

%e 6 |_ _ _| | |_ _ _| | |

%e 7 .... 7 ....................................|_ _ _ _ _| |_ _ _ _ _| |

%e 8 |_ _| | |

%e 9 |_ _ _ _| |

%e 10 |_ _ _| |

%e 11 .. 11 .................................................|_ _ _ _ _ _|

%e .

%e Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.

%e Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.

%e The total number of parts equals the sum of largest parts.

%e Note that every diagram contains all previous diagrams.

%e An infinite diagram is a table of all partitions of all positive integers.

%p with(combinat): seq(4*numbpart(n),n=0..50); # _Muniru A Asiru_, Jul 10 2018

%t 4*PartitionsP[Range[0,50]] (* _Harvey P. Dale_, Dec 05 2023 *)

%o (GAP) List([0..50],n->4*NrPartitions(n)); # _Muniru A Asiru_, Jul 10 2018

%o (PARI) a(n) = 4*numbpart(n); \\ _Michel Marcus_, Jul 15 2018

%o (Python)

%o from sympy.ntheory import npartitions

%o def a(n): return 4*npartitions(n)

%o print([a(n) for n in range(51)]) # _Michael S. Branicky_, Apr 04 2021

%Y k times partition numbers: A000041 (k=1), A139582 (k=2), A299473 (k=3), this sequence (k=4).

%Y Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

%K nonn

%O 0,1

%A _Omar E. Pol_, Feb 10 2018