A064064 - OEIS

%I #40 Feb 25 2024 09:10:18

%S 1,2,3,4,4,5,6,5,6,7,8,5,6,7,8,8,6,7,8,9,9,10,7,8,9,10,10,11,12,6,7,8,

%T 9,9,10,11,10,7,8,9,10,10,11,12,11,12,8,9,10,11,11,12,13,12,13,14,9,

%U 10,11,12,12,13,14,13,14,15,16,6,7,8,9,9,10,11,10,11,12,13,10,7,8,9,10,10

%N n-th step is to add a(n) to each previous number a(k) (including itself, i.e., k <= n) to produce n+1 more terms of the sequence, starting with a(0)=1.

%C Each number eventually appears A001190 times (binary rooted trees can be constructed by combining earlier trees in a similar manner with the n-th tree having a(n) endpoints).

%C The number of leaves in the tree of rank n+1 according to the Colijn-Plazzotta ranking scheme for unlabeled binary rooted trees. - _Noah A Rosenberg_, Jun 14 2022

%H Rémy Sigrist, <a href="/A064064/b064064.txt">Table of n, a(n) for n = 0..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, volume 67, number 1, January 2018, pages 113-126, with section 2.3 number of tips v_a(1) = a(n) for tree R_a = n+1.

%H N. A. Rosenberg, <a href="https://doi.org/10.1016/j.dam.2020.11.021">On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees</a>, Discr. Appl. Math. 291 (2021), 88-98.

%F a(0) = 1 and a(n+1) = a(A002262(n)) + a(A003056(n)) for any n >= 0. - _Rémy Sigrist_, Aug 07 2017

%F a(n) = (A064002(n+1) + 1)/2. - _Kevin Ryde_, Jul 22 2022

%e Start with (1). So after initial step we have (*1*, 1+1 = 2), then (1, *2*, 1+2 = 3, 2+2 = 4), then (1, 2, *3*, 4, 1+3 = 4, 2+3 = 5, 3+3 = 6), then (1, 2, 3, *4*, 4, 5, 6, 1+4 = 5, 2+4 = 6, 3+4 = 7, 4+4 = 8), then (1, 2, 3, 4, *4*, 5, 6, 5, 6, 7, 8, 1+4 = 5, 2+4 = 6, 3+4 = 7, 4+4 = 8, 4+4 = 8), etc.

%t a[0]=1; a[n_]:=With[{s = Floor[(Sqrt[8*n - 7] - 1)/2]}, a[s] + a[n - s*(s + 1)/2 - 1]]; Array[a,84,0] (* _Harry Richman_, Feb 24 2024 *)

%o (PARI) a(n) = if (n==0, return (1), return (a(A002262(n-1))+a(A003056(n-1)))) \\ _Rémy Sigrist_, Aug 07 2017

%Y Cf. A001190, A064065, A064066, A064067.

%Y Cf. A002262, A003056, A064002.

%K nonn,easy

%O 0,2

%A _Henry Bottomley_, Aug 31 2001