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A220517
First differences of A225600. Also A141285 and A194446 interleaved.
17
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
1,3
COMMENTS
Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.
FORMULA
a(2n-1) = A141285(n); a(2n) = A194446(n), n >= 1
EXAMPLE
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
.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Feb 07 2013
STATUS
approved