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%I #16 Feb 06 2024 11:32:09
%S 1,3,5,7,7,9,11,9,11,13,15,9,11,13,15,15,11,13,15,17,17,19,13,15,17,
%T 19,19,21,23,11,13,15,17,17,19,21,19,13,15,17,19,19,21,23,21,23,15,17,
%U 19,21,21,23,25,23,25,27,17,19,21,23,23,25,27,25,27,29,31,11,13,15,17,17
%N List pairs (i,j) with 1 <= i <= j in colexicographic order: (1,1), (1,2), (2,2), (1,3), (2,3), (3,3), (1,4), ... Let a(1) = 1. Then for n>=2 if the (n-1)-st pair is (i,j) then a(n) = a(i) + a(j) + 1.
%C All entries are odd. There are A001190(n) occurrences of 2n-1 in this sequence.
%C a(n) is the number of vertices in the rooted binary tree (every vertex 0 or 2 children) with Colijn-Plazzotta tree number n. - _Kevin Ryde_, Jul 25 2022
%H Kevin Ryde, <a href="/A064002/b064002.txt">Table of n, a(n) for n = 1..10000</a>
%H C. Colijn and G. Plazzotta, <a href="https://doi.org/10.1093/sysbio/syx046">A Metric on Phylogenetic Tree Shapes</a>, Systematic Biology, 67 (1) (2018), 113-126.
%H Kevin Ryde, <a href="/A064002/a064002.gp.txt">PARI/GP Code</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lexicographic_order#Colexicographic_order">Colexicographic order</a>
%F a(n) = 2*A064064(n-1) - 1. - _Kevin Ryde_, Jul 25 2022
%e a(2) = a(1)+a(1)+1 = 3,
%e a(3) = a(1)+a(2)+1 = 5,
%e a(4) = a(2)+a(2)+1 = 7,
%e a(5) = a(1)+a(3)+1 = 7, ...
%o (PARI) See links.
%o (Python)
%o from itertools import count, islice
%o def bgen(): yield from ((i, j) for j in count(1) for i in range(1, j+1))
%o def agen():
%o a, g = [None, 1], bgen()
%o for n in count(2):
%o yield a[-1];
%o i, j = next(g)
%o a.append(a[i] + a[j] + 1)
%o print(list(islice(agen(), 72))) # _Michael S. Branicky_, Jul 25 2022
%Y Cf. A001190, A064064.
%K easy,nonn
%O 1,2
%A Claude Lenormand (claude.lenormand(AT)free.fr), Sep 14 2001
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