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

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, 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, 6, 4, 5, 1, 4, 1, 8, 7, 4, 1, 7, 2, 6, 1, 5, 1, 10, 42

Offset: 1

Author

Omar E. Pol, Feb 07 2013

Keywords

• nonn
• tabf

Comments

Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.

Examples

			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.
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;
2,1,4,2,3,1,6,4,5,1,4,1,8,22;
3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30;
2,1,4,2,3,1,6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42;
.
Illustration of the first seven rows of triangle as a minimalist diagram of regions of the set of partitions of 7:
.      _ _ _ _ _ _ _
. 15   _ _ _ _      |
.      _ _ _ _|_    |
.      _ _ _    |   |
.      _ _ _|_ _|_  |
. 11   _ _ _      | |
.      _ _ _|_    | |
.      _ _    |   | |
.      _ _|_ _|_  | |
.  7   _ _ _    | | |
.      _ _ _|_  | | |
.  5   _ _    | | | |
.      _ _|_  | | | |
.  3   _ _  | | | | |
.  2   _  | | | | | |
.  1    | | | | | | |
.
.      1 2 3 4 5 6 7
.
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:
.
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
. 0,2,  6,   12,         24,             40... = A211978
.  1, 4,   9,       19,           33... = A179862
.
		

Links

• Omar E. Pol, Visualization of regions in a minimalist diagram for A006128
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A000041, A006128, A135010, A138137, A141285, A179862, A186114, A186412, A187219, A194446, A206437, A211978, A220517, A225600, A225610.

Formula

a(2n-1) = A141285(n); a(2n) = A194446(n), n >= 1