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Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A211987 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral. 2 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 5, 6, 3, 3, 4, 2, 2, 2, 2, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 5, 1, 1, 6, 1, 3, 3, 1, 4, 2, 1 (list; graph; refs; listen; history; text; internal format) OFFSET 1,2 COMMENTS In order to construct this sequence we use the following rules: - Consider the partitions of positive integers. - For each positive integer its shells must be arranged in a spiral. - The sequence lists one spiral for each positive integer. - If the integer j is odd then the last composition listed of each spiral is j. - If the integer j is even then the first composition listed of each spiral is j. This sequence represents a three-dimensional structure in which each column contains parts of the same size. LINKS Table of n, a(n) for n=1..86. Omar E. Pol, Illustration of the transformation of the partitions of 7 (fig. A) into the rows 30..44 (fig. D) EXAMPLE ---------------------------------------------- . Expanded Geometric Compositions arrangement model ---------------------------------------------- 1; 1; |*| ---------------------------------------------- 2; . 2; |* *| 1,1; 1,1; |o|*| ---------------------------------------------- 1,2; 1,. 2; |*|o o| 1,1,1; 1,1,1; |*|o|o| 3; 3 . .; |* * *| ---------------------------------------------- 4,; . . . 4; |* * * *| 2,2; . 2,. 2; |* *|* *| 1,2,1; 1,. 2,1; |o|o o|*| 1,1,1,1,; 1,1,1,1; |o|o|o|*| 3,1; 3 . .,1; |o o o|*| ---------------------------------------------- 1,4; 1,. . . 4; |*|o o o o| 1,2,2; 1,. 2,. 2; |*|o o|o o| 1,1,2,1; 1,1,. 2,1; |*|o|o o|o| 1,1,1,1,1; 1,1,1,1,1; |*|o|o|o|o| 1,3,1; 1,3 . .,1; |*|o o o|o| 2,3; 2 .,3 . .; |* *|* * *| 5; 5 . . . .; |* * * * *| ---------------------------------------------- 6; . . . . . 6; |* * * * * *| 3,3; . . 3,. . 3; |* * *|* * *| 4,2; . . . 4,. 2; |* * * *|* *| 2,2,2; . 2,. 2,. 2; |* *|* *|* *| 1,4,1; 1,. . . 4,1; |o|o o o o|*| 1,2,2,1; 1,. 2,. 2,1; |o|o o|o o|*| 1,1,2,1,1; 1,1,. 2,1,1; |o|o|o o|o|*| 1,1,1,1,1,1; 1,1,1,1,1,1; |o|o|o|o|o|*| 1,3,1,1; 1,3 . .,1,1; |o|o o o|o|*| 2,3,1; 2 .,3 . .,1; |o o|o o o|*| 5,1; 5 . . . .,1; |o o o o o|*| ---------------------------------------------- Note that * is a unitary element of every part of the last section of j. CROSSREFS Rows sums give A036042, n>=1. Mirror of A211988. Other spiral versions are A211985, A211986, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999. Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437. Sequence in context: A170981 A161096 A214247 * A165983 A300719 A341998 Adjacent sequences: A211984 A211985 A211986 * A211988 A211989 A211990 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Aug 18 2012 STATUS approved

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Last modified August 26 19:22 EDT 2024. Contains 375462 sequences. (Running on oeis4.)