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 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. 17
 1, 4, 10, 18, 33, 52, 87, 130, 202, 295, 436, 617, 887, 1226, 1709, 2327, 3173, 4244, 5691, 7505, 9907, 12917, 16822, 21690, 27947, 35685, 45506, 57625, 72836, 91500, 114760, 143143, 178235, 220908, 273268, 336670, 414041, 507298, 620455, 756398, 920470 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS Omar E. Pol, Illustration of the seven regions of 5 N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS FORMULA a(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n). EXAMPLE 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. .                               Diagram of regions Partitions of 7                 and partitions of 7 .                                   _ _ _ _ _ _ _ 7                               15 |_ _ _ _      | 4 + 3                              |_ _ _ _|_    | 5 + 2                              |_ _ _    |   | 3 + 2 + 2                          |_ _ _|_ _|_  | 6 + 1                           11 |_ _ _      | | 3 + 3 + 1                          |_ _ _|_    | | 4 + 2 + 1                          |_ _    |   | | 2 + 2 + 2 + 1                      |_ _|_ _|_  | | 5 + 1 + 1                        7 |_ _ _    | | | 3 + 2 + 1 + 1                      |_ _ _|_  | | | 4 + 1 + 1 + 1                    5 |_ _    | | | | 2 + 2 + 1 + 1 + 1                  |_ _|_  | | | | 3 + 1 + 1 + 1 + 1                3 |_ _  | | | | | 2 + 1 + 1 + 1 + 1 + 1            2 |_  | | | | | | 1 + 1 + 1 + 1 + 1 + 1 + 1        1 |_|_|_|_|_|_|_| . .                                   1 2 3 4 5 6 7 . 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: .                                         _ _ _ _ _ _ .                                        |_ _ _      | .                                        |_ _ _|_    | .                                        |_ _    |   | .                             _ _ _ _ _  |_ _|_ _|_  | .                            |_ _ _    | |_ _ _    | | .                   _ _ _ _  |_ _ _|_  | |_ _ _|_  | | .                  |_ _    | |_ _    | | |_ _    | | | .           _ _ _  |_ _|_  | |_ _|_  | | |_ _|_  | | | .     _ _  |_ _  | |_ _  | | |_ _  | | | |_ _  | | | | . _  |_  | |_  | | |_  | | | |_  | | | | |_  | | | | | .|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_| . . 4    10     18       33         52          87 CROSSREFS Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600. Sequence in context: A009867 A019455 A073839 * A009921 A050188 A062361 Adjacent sequences:  A225607 A225608 A225609 * A225611 A225612 A225613 KEYWORD nonn AUTHOR Omar E. Pol, Jul 29 2013 STATUS approved

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Last modified July 11 20:03 EDT 2020. Contains 335652 sequences. (Running on oeis4.)