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 A229946 Height of the peaks and the valleys in the Dyck path whose j-th ascending line segment has A141285(j) steps and whose j-th descending line segment has A194446(j) steps. 3
 0, 1, 0, 2, 0, 3, 0, 2, 1, 5, 0, 3, 2, 7, 0, 2, 1, 5, 3, 6, 5, 11, 0, 3, 2, 7, 5, 9, 8, 15, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 0, 3, 2, 7, 5, 9, 8, 15, 11, 14, 13, 19, 17, 22, 21, 30, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 15, 19, 18, 25, 23, 29, 28, 33, 32, 42, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also 0 together the alternating sums of A220517. The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region. For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y].  See below: . .  j     Diagram 1         Partitions           Diagram 2 .      _ _ _ _ _ _ _                          _ _ _ _ _ _ _ . 15  |_ _ _ _      |      7                  _ _ _ _      | . 14  |_ _ _ _|_    |      4+3                _ _ _ _|_    | . 13  |_ _ _    |   |      5+2                _ _ _    |   | . 12  |_ _ _|_ _|_  |      3+2+2              _ _ _|_ _|_  | . 11  |_ _ _      | |      6+1                _ _ _      | | . 10  |_ _ _|_    | |      3+3+1              _ _ _|_    | | .  9  |_ _    |   | |      4+2+1              _ _    |   | | .  8  |_ _|_ _|_  | |      2+2+2+1            _ _|_ _|_  | | .  7  |_ _ _    | | |      5+1+1              _ _ _    | | | .  6  |_ _ _|_  | | |      3+2+1+1            _ _ _|_  | | | .  5  |_ _    | | | |      4+1+1+1            _ _    | | | | .  4  |_ _|_  | | | |      2+2+1+1+1          _ _|_  | | | | .  3  |_ _  | | | | |      3+1+1+1+1          _ _  | | | | | .  2  |_  | | | | | |      2+1+1+1+1+1        _  | | | | | | .  1  |_|_|_|_|_|_|_|      1+1+1+1+1+1+1       | | | | | | | . .      1 2 3 4 5 6 7 . The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1. Also the diagram has the property that it can be transformed in a Dyck path (see example). The height of the peaks and the valleys of the infinite Dyck path give this sequence. Q: Is this Dyck path a fractal? LINKS Omar E. Pol, Visualization of regions in a diagram for A006128 FORMULA a(0) = 0; a(n) = a(n-1) + (-1)^(n-1)*A220517(n), n >= 1. EXAMPLE Illustration of initial terms (n = 0..21): .                                                             11 .                                                             / .                                                            / .                                                           / .                                   7                      / .                                   /\                 6  / .                     5            /  \           5    /\/ .                     /\          /    \          /\  / 5 .           3        /  \     3  /      \        /  \/ .      2    /\   2  /    \    /\/        \   2  /   3 .   1  /\  /  \  /\/      \  / 2          \  /\/ .   /\/  \/    \/ 1        \/              \/ 1 .  0 0   0     0           0               0 . Note that the k-th largest peak between two valleys at height 0 is also A000041(k) and the next term is always 0. . Written as an irregular triangle in which row k has length 2*A187219(k), k >= 1, the sequence begins: 0,1; 0,2; 0,3; 0,2,1,5; 0,3,2,7; 0,2,1,5,3,6,5,11; 0,3,2,7,5,9,8,15; 0,2,1,5,3,6,5,11,7,12,11,15,14,22; 0,3,2,7,5,9,8,15,11,14,13,19,17,22,21,30; 0,2,1,5,3,6,5,11,7,12,11,15,14,22,15,19,18,25,23,29,28,33,32,42; ... CROSSREFS Column 1 is A000004. Right border gives A000041 for the positive integers. Cf. A006128, A135010, A138137, A139582, A141285, A186412, A187219, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610. Sequence in context: A027640 A194666 A325799 * A127460 A274021 A303711 Adjacent sequences:  A229943 A229944 A229945 * A229947 A229948 A229949 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Nov 03 2013 STATUS approved

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Last modified February 24 02:48 EST 2020. Contains 332195 sequences. (Running on oeis4.)