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%I #16 Jul 18 2022 22:48:15
%S 1,2,1,1,3,1,1,1,1,2,4,2,2,1,2,1,1,1,1,1,3,1,5,2,3,1,3,1,1,1,1,1,1,1,
%T 1,2,1,1,2,2,1,4,6,3,3,4,2,2,2,2,1,4,1,1,2,2,1,1,1,2,1,1,1,1,1,1,1,1,
%U 1,3,1,1,2,3,1,5,1,7,3,4,2,5,2,2,3,1,5,1,1,2,3,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,4,1,1,2,2,2,1,4,2,1,3,3,1,6
%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 we use the same spiral of A211995.
%C - If the integer j is even then the first composition listed of each spiral is j.
%e --------------------------------------------
%e . Expanded Geometric
%e Compositions arrangement model
%e --------------------------------------------
%e 1; 1; |*|
%e --------------------------------------------
%e 2; . 2; |* *|
%e 1,1; 1,1; |o|*|
%e --------------------------------------------
%e 3; 3 . .; |* * *|
%e 1,1,1; 1,1,1; |*|o|o|
%e 1,2; 1,. 2; |*|o o|
%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 5; 5 . . . .; |* * * * *|
%e 2,3; 2 .,3 . .; |* *|* * *|
%e 1,3,1; 1,3 . .,1; |*|o o o|o|
%e 1,1,1,1,1; 1,1,1,1,1; |*|o|o|o|o|
%e 1,1,2,1; 1,1,. 2,1; |*|o|o o|o|
%e 1,2,2; 1,. 2,. 2; |*|o o|o o|
%e 1,4; 1,. . . 4; |*|o o o o|
%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 A211986. Other spiral versions are A211987, A211988, 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 19 2012