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 A225610 #37 Aug 05 2013 03:39:35 %S A225610 1,4,10,18,33,52,87,130,202,295,436,617,887,1226,1709,2327,3173,4244, %T A225610 5691,7505,9907,12917,16822,21690,27947,35685,45506,57625,72836,91500, %U A225610 114760,143143,178235,220908,273268,336670,414041,507298,620455,756398,920470 %N A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n. %C A225610 a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600. %H A225610 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a> %H A225610 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a> %H A225610 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %F A225610 a(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n). %e A225610 For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130. %e A225610 . Diagram of regions %e A225610 Partitions of 7 and partitions of 7 %e A225610 . _ _ _ _ _ _ _ %e A225610 7 15 |_ _ _ _ | %e A225610 4 + 3 |_ _ _ _|_ | %e A225610 5 + 2 |_ _ _ | | %e A225610 3 + 2 + 2 |_ _ _|_ _|_ | %e A225610 6 + 1 11 |_ _ _ | | %e A225610 3 + 3 + 1 |_ _ _|_ | | %e A225610 4 + 2 + 1 |_ _ | | | %e A225610 2 + 2 + 2 + 1 |_ _|_ _|_ | | %e A225610 5 + 1 + 1 7 |_ _ _ | | | %e A225610 3 + 2 + 1 + 1 |_ _ _|_ | | | %e A225610 4 + 1 + 1 + 1 5 |_ _ | | | | %e A225610 2 + 2 + 1 + 1 + 1 |_ _|_ | | | | %e A225610 3 + 1 + 1 + 1 + 1 3 |_ _ | | | | | %e A225610 2 + 1 + 1 + 1 + 1 + 1 2 |_ | | | | | | %e A225610 1 + 1 + 1 + 1 + 1 + 1 + 1 1 |_|_|_|_|_|_|_| %e A225610 . %e A225610 . 1 2 3 4 5 6 7 %e A225610 . %e A225610 Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6: %e A225610 . _ _ _ _ _ _ %e A225610 . |_ _ _ | %e A225610 . |_ _ _|_ | %e A225610 . |_ _ | | %e A225610 . _ _ _ _ _ |_ _|_ _|_ | %e A225610 . |_ _ _ | |_ _ _ | | %e A225610 . _ _ _ _ |_ _ _|_ | |_ _ _|_ | | %e A225610 . |_ _ | |_ _ | | |_ _ | | | %e A225610 . _ _ _ |_ _|_ | |_ _|_ | | |_ _|_ | | | %e A225610 . _ _ |_ _ | |_ _ | | |_ _ | | | |_ _ | | | | %e A225610 . _ |_ | |_ | | |_ | | | |_ | | | | |_ | | | | | %e A225610 .|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_| %e A225610 . %e A225610 . 4 10 18 33 52 87 %Y A225610 Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600. %K A225610 nonn %O A225610 0,2 %A A225610 _Omar E. Pol_, Jul 29 2013