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A064002 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.

%I A064002 #17 Dec 19 2024 11:46:19
%S A064002 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 A064002 19,19,21,23,11,13,15,17,17,19,21,19,13,15,17,19,19,21,23,21,23,15,17,
%U A064002 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 A064002 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 A064002 All entries are odd. There are A001190(n) occurrences of 2n-1 in this sequence.
%C A064002 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 A064002 Kevin Ryde, <a href="/A064002/b064002.txt">Table of n, a(n) for n = 1..10000</a>
%H A064002 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 A064002 Kevin Ryde, <a href="/A064002/a064002.gp.txt">PARI/GP Code</a>
%H A064002 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lexicographic_order#Colexicographic_order">Colexicographic order</a>
%F A064002 a(n) = 2*A064064(n-1) - 1. - _Kevin Ryde_, Jul 25 2022
%e A064002 a(2) = a(1)+a(1)+1 = 3,
%e A064002 a(3) = a(1)+a(2)+1 = 5,
%e A064002 a(4) = a(2)+a(2)+1 = 7,
%e A064002 a(5) = a(1)+a(3)+1 = 7, ...
%o A064002 (PARI) \\ See links.
%o A064002 (Python)
%o A064002 from itertools import count, islice
%o A064002 def bgen(): yield from ((i, j) for j in count(1) for i in range(1, j+1))
%o A064002 def agen():
%o A064002     a, g = [None, 1], bgen()
%o A064002     for n in count(2):
%o A064002         yield a[-1];
%o A064002         i, j = next(g)
%o A064002         a.append(a[i] + a[j] + 1)
%o A064002 print(list(islice(agen(), 72))) # _Michael S. Branicky_, Jul 25 2022
%Y A064002 Cf. A001190, A064064.
%K A064002 easy,nonn
%O A064002 1,2
%A A064002 Claude Lenormand (claude.lenormand(AT)free.fr), Sep 14 2001