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%I #8 Jan 18 2014 16:21:49
%S 2,4,6,12,16,30,38,64,84,128,166,248,314,448,576,790,1004,1358,1708,
%T 2264,2844,3694,4614,5936,7354,9342,11544,14502,17816,22220,27144,
%U 33584,40878,50192,60828,74276,89596,108778,130772,157918,189116,227374
%N Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.
%C Also first differences of A211978.
%F a(n) = 2*(A006128(n) - A006128(n-1)) = 2*A138137(n).
%e Illustration of initial terms as a dissection of a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
%e . _ _ _ _ _ _
%e . _ _ _ |
%e . _ _ _|_ |
%e . _ _ | |
%e . _ _ _ _ _ | | |
%e . _ _ _ | |
%e . _ _ _ _ | | |
%e . _ _ | | |
%e . _ _ _ | | | |
%e . _ _ | | | |
%e . _ | | | | |
%e . | | | | | |
%e .
%e . 2 4 6 12 16 30
%e .
%e Also using the elements from the above diagram we can draw an infinite 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 n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
%e 7..................................
%e . /\
%e 5.................... / \ /\
%e . /\ / \ /\ /
%e 3.......... / \ / \ / \/
%e 2..... /\ / \ /\/ \ /
%e 1.. /\ / \ /\/ \ / \ /\/
%e 0 /\/ \/ \/ \/ \/
%e . 2, 4, 6, 12, 16,...
%e .
%Y Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228109, A228110, A229946.
%K nonn
%O 1,1
%A _Omar E. Pol_, Jan 14 2014
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