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 A352446 #47 Oct 23 2024 09:55:27
%S A352446 0,0,2,0,2,0,2,0,2,4,2,0,2,4,2,0,2,4,2,0,8,4,2,0,2,4,8,0,2,4,2,0,8,4,
%T A352446 2,8,2,4,8,0,2,4,2,8,8,4,2,0,2,4,8,8,2,4,12,0,8,4,2,8,2,4,8,0,12,4,2,
%U A352446 8,8,4,2,0,2,4,18,8,2,16,2,0,8,4,2,8,12,4,8,0,2,16,2
%N A352446 Total number of parts in all partitions of n into an even number of consecutive parts.
%F A352446 a(n) = A204217(n) - A341309(n), conjectured.
%F A352446 Conjecture: G.f.: Sum_{k>=1} 2*k*x^(k*(2*k+1))/(1-x^(2*k)). - _Vaclav Kotesovec_, Oct 23 2024
%e A352446 For n = 21 the partitions of 21 into an even number of consecutive parts are [11, 10] and [6, 5, 4, 3, 2, 1]. The total number of parts in these two partitions is equal to 2 + 6 = 8, so a(21) = 8.
%e A352446 On the other hand consider the diagram below which is formed by the even-indexed staircase walks from the diagram of A286000.
%e A352446 The diagram is infinite and we have that:
%e A352446 The m-th staircase walk starts at row A014105(m).
%e A352446 The number of horizontal line segment in the n-th row equals A131576(n), the number of partitions of n into an even number of consecutive parts.
%e A352446 a(n) is the total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the n-th level of the diagram.
%e A352446 ---------------------------------------------
%e A352446 n a(n) Diagram
%e A352446 ---------------------------------------------
%e A352446 1 0
%e A352446 2 0 _
%e A352446 3 2 |2
%e A352446 4 0 _|1
%e A352446 5 2 |3
%e A352446 6 0 _|2
%e A352446 7 2 |4
%e A352446 8 0 _|3
%e A352446 9 2 |5 _
%e A352446 10 4 _|4 |4
%e A352446 11 2 |6 |3
%e A352446 12 0 _|5 |2
%e A352446 13 2 |7 _|1
%e A352446 14 4 _|6 |5
%e A352446 15 2 |8 |4
%e A352446 16 0 _|7 |3
%e A352446 17 2 |9 _|2
%e A352446 18 4 _|8 |6
%e A352446 19 2 |10 |5
%e A352446 20 0 _|9 |4 _
%e A352446 21 8 |11 _|3 |6
%e A352446 22 4 _|10 |7 |5
%e A352446 23 2 |12 |6 |4
%e A352446 24 0 _|11 |5 |3
%e A352446 25 2 |13 _|4 |2
%e A352446 26 4 _|12 |8 _|1
%e A352446 27 8 |14 |7 |7
%e A352446 28 0 |13 |6 |6
%e A352446 ...
%e A352446 For n = 21 the number of horizontal line segment in the 21th row of the diagram equals A131576(21) = 2, the number of partitions of 21 into an even number of consecutive parts.
%e A352446 The total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the 21-th level of the diagram is equal to 2 + 6 = 8, so a(21) = 8.
%Y A352446 For more information about the diagram see A286000 and A237593.
%Y A352446 Cf. A014105, A131576, A204217, A299765, A341309, A352505.
%K A352446 nonn
%O A352446 1,3
%A A352446 _Omar E. Pol_, Mar 16 2022