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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.
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%I #24 Jul 18 2022 22:48:07

%S 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,

%T 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,

%U 1,3,1,1,2,3,1,5,1,1,6,1,3,3,1,4,2,1

%N 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.

%C In order to construct this sequence we use the following rules:

%C - Consider the partitions of positive integers.

%C - For each positive integer its shells must be arranged in a spiral.

%C - The sequence lists one spiral for each positive integer.

%C - If the integer j is odd then the last composition listed of each spiral is j.

%C - If the integer j is even then the first composition listed of each spiral is j.

%C This sequence represents a three-dimensional structure in which each column contains parts of the same size.

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa403.jpg">Illustration of the transformation of the partitions of 7 (fig. A) into the rows 30..44 (fig. D)</a>

%e ----------------------------------------------

%e . Expanded Geometric

%e Compositions arrangement model

%e ----------------------------------------------

%e 1; 1; |*|

%e ----------------------------------------------

%e 2; . 2; |* *|

%e 1,1; 1,1; |o|*|

%e ----------------------------------------------

%e 1,2; 1,. 2; |*|o o|

%e 1,1,1; 1,1,1; |*|o|o|

%e 3; 3 . .; |* * *|

%e ----------------------------------------------

%e 4,; . . . 4; |* * * *|

%e 2,2; . 2,. 2; |* *|* *|

%e 1,2,1; 1,. 2,1; |o|o o|*|

%e 1,1,1,1,; 1,1,1,1; |o|o|o|*|

%e 3,1; 3 . .,1; |o o o|*|

%e ----------------------------------------------

%e 1,4; 1,. . . 4; |*|o o o o|

%e 1,2,2; 1,. 2,. 2; |*|o o|o o|

%e 1,1,2,1; 1,1,. 2,1; |*|o|o o|o|

%e 1,1,1,1,1; 1,1,1,1,1; |*|o|o|o|o|

%e 1,3,1; 1,3 . .,1; |*|o o o|o|

%e 2,3; 2 .,3 . .; |* *|* * *|

%e 5; 5 . . . .; |* * * * *|

%e ----------------------------------------------

%e 6; . . . . . 6; |* * * * * *|

%e 3,3; . . 3,. . 3; |* * *|* * *|

%e 4,2; . . . 4,. 2; |* * * *|* *|

%e 2,2,2; . 2,. 2,. 2; |* *|* *|* *|

%e 1,4,1; 1,. . . 4,1; |o|o o o o|*|

%e 1,2,2,1; 1,. 2,. 2,1; |o|o o|o o|*|

%e 1,1,2,1,1; 1,1,. 2,1,1; |o|o|o o|o|*|

%e 1,1,1,1,1,1; 1,1,1,1,1,1; |o|o|o|o|o|*|

%e 1,3,1,1; 1,3 . .,1,1; |o|o o o|o|*|

%e 2,3,1; 2 .,3 . .,1; |o o|o o o|*|

%e 5,1; 5 . . . .,1; |o o o o o|*|

%e ----------------------------------------------

%e Note that * is a unitary element of every part of the last section of j.

%Y Rows sums give A036042, n>=1.

%Y Mirror of A211988. Other spiral versions are A211985, A211986, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.

%Y Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Aug 18 2012