This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A299474 #39 Dec 05 2023 17:01:13 %S A299474 4,4,8,12,20,28,44,60,88,120,168,224,308,404,540,704,924,1188,1540, %T A299474 1960,2508,3168,4008,5020,6300,7832,9744,12040,14872,18260,22416, %U A299474 27368,33396,40572,49240,59532,71908,86548,104060,124740,149352,178332,212696,253044,300700,356536,422232,499016,589092,694100,816904 %N A299474 a(n) = 4*p(n), where p(n) is the number of partitions of n. %C A299474 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 A299474 Shawn A. Broyles, <a href="/A299474/b299474.txt">Table of n, a(n) for n = 0..1000</a> %F A299474 a(n) = 4*A000041(n) = 2*A139582(n). %F A299474 a(n) = A000041(n) + A299475(n) - 1, n >= 1 (Euler's formula). %F A299474 a(n) = A000041(n) + A299473(n). - _Omar E. Pol_, Aug 11 2018 %e A299474 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 A299474 -------------------------------------------------------------------------------- %e A299474 n ........: 1 2 3 4 5 6 (stage) %e A299474 a(n)......: 4 8 12 20 28 44 (edges) %e A299474 A299475(n): 4 7 10 16 22 34 (vertices) %e A299474 A000041(n): 1 2 3 5 7 11 (regions) %e A299474 -------------------------------------------------------------------------------- %e A299474 r p(n) %e A299474 -------------------------------------------------------------------------------- %e A299474 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A299474 1 .... 1 ....|_| |_| | |_| | | |_| | | | |_| | | | | |_| | | | | | %e A299474 2 .... 2 .........|_ _| |_ _| | |_ _| | | |_ _| | | | |_ _| | | | | %e A299474 3 .... 3 ................|_ _ _| |_ _ _| | |_ _ _| | | |_ _ _| | | | %e A299474 4 |_ _| | |_ _| | | |_ _| | | | %e A299474 5 .... 5 .........................|_ _ _ _| |_ _ _ _| | |_ _ _ _| | | %e A299474 6 |_ _ _| | |_ _ _| | | %e A299474 7 .... 7 ....................................|_ _ _ _ _| |_ _ _ _ _| | %e A299474 8 |_ _| | | %e A299474 9 |_ _ _ _| | %e A299474 10 |_ _ _| | %e A299474 11 .. 11 .................................................|_ _ _ _ _ _| %e A299474 . %e A299474 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 A299474 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 A299474 The total number of parts equals the sum of largest parts. %e A299474 Note that every diagram contains all previous diagrams. %e A299474 An infinite diagram is a table of all partitions of all positive integers. %p A299474 with(combinat): seq(4*numbpart(n),n=0..50); # _Muniru A Asiru_, Jul 10 2018 %t A299474 4*PartitionsP[Range[0,50]] (* _Harvey P. Dale_, Dec 05 2023 *) %o A299474 (GAP) List([0..50],n->4*NrPartitions(n)); # _Muniru A Asiru_, Jul 10 2018 %o A299474 (PARI) a(n) = 4*numbpart(n); \\ _Michel Marcus_, Jul 15 2018 %o A299474 (Python) %o A299474 from sympy.ntheory import npartitions %o A299474 def a(n): return 4*npartitions(n) %o A299474 print([a(n) for n in range(51)]) # _Michael S. Branicky_, Apr 04 2021 %Y A299474 k times partition numbers: A000041 (k=1), A139582 (k=2), A299473 (k=3), this sequence (k=4). %Y A299474 Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475. %K A299474 nonn %O A299474 0,1 %A A299474 _Omar E. Pol_, Feb 10 2018