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A220517 First differences of A225600. Also A141285 and A194446 interleaved.

%I A220517 #29 Nov 04 2013 18:06:27
%S A220517 1,1,2,2,3,3,2,1,4,5,3,1,5,7,2,1,4,2,3,1,6,11,3,1,5,2,4,1,7,15,2,1,4,
%T A220517 2,3,1,6,4,5,1,4,1,8,22,3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30,2,1,4,2,3,1,
%U A220517 6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42
%N A220517 First differences of A225600. Also A141285 and A194446 interleaved.
%C A220517 Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.
%H A220517 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a minimalist diagram for A006128</a>
%H A220517 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A220517 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A220517 a(2n-1) = A141285(n); a(2n) = A194446(n), n >= 1
%e A220517 Written as an irregular triangle in which row n has length 2*A187219(n) we can see that the right border gives A000041 and the previous term of the last term in row n is n.
%e A220517 1,1;
%e A220517 2,2;
%e A220517 3,3;
%e A220517 2,1,4,5;
%e A220517 3,1,5,7;
%e A220517 2,1,4,2,3,1,6,11;
%e A220517 3,1,5,2,4,1,7,15;
%e A220517 2,1,4,2,3,1,6,4,5,1,4,1,8,22;
%e A220517 3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30;
%e A220517 2,1,4,2,3,1,6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42;
%e A220517 .
%e A220517 Illustration of the first seven rows of triangle as a minimalist diagram of regions of the set of partitions of 7:
%e A220517 .      _ _ _ _ _ _ _
%e A220517 . 15   _ _ _ _      |
%e A220517 .      _ _ _ _|_    |
%e A220517 .      _ _ _    |   |
%e A220517 .      _ _ _|_ _|_  |
%e A220517 . 11   _ _ _      | |
%e A220517 .      _ _ _|_    | |
%e A220517 .      _ _    |   | |
%e A220517 .      _ _|_ _|_  | |
%e A220517 .  7   _ _ _    | | |
%e A220517 .      _ _ _|_  | | |
%e A220517 .  5   _ _    | | | |
%e A220517 .      _ _|_  | | | |
%e A220517 .  3   _ _  | | | | |
%e A220517 .  2   _  | | | | | |
%e A220517 .  1    | | | | | | |
%e A220517 .
%e A220517 .      1 2 3 4 5 6 7
%e A220517 .
%e A220517 Also using the elements of this diagram we can draw a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See below:
%e A220517 .
%e A220517 7..................................
%e A220517 .                                 /\
%e A220517 5....................            /  \                /\
%e A220517 .                   /\          /    \          /\  /
%e A220517 3..........        /  \        /      \        /  \/
%e A220517 2.....    /\      /    \    /\/        \      /
%e A220517 1..  /\  /  \  /\/      \  /            \  /\/
%e A220517 0 /\/  \/    \/          \/              \/
%e A220517 . 0,2,  6,   12,         24,             40... = A211978
%e A220517 .  1, 4,   9,       19,           33... = A179862
%e A220517 .
%Y A220517 Cf. A000041, A006128, A135010, A138137, A141285, A179862, A186114, A186412, A187219, A194446, A206437, A211978, A220517, A225600, A225610.
%K A220517 nonn,tabf
%O A220517 1,3
%A A220517 _Omar E. Pol_, Feb 07 2013