

A064064


nth 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.


7



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, 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, 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
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OFFSET

0,2


COMMENTS

Each number eventually appears A001190 times (binary rooted trees can be constructed by combining earlier trees in a similar manner with the nth tree having a(n) endpoints).
The number of leaves in the tree of rank n+1 according to the ColijnPlazzotta ranking scheme for unlabeled binary rooted trees.  Noah A Rosenberg, Jun 14 2022


LINKS

C. Colijn and G. Plazzotta, A Metric on Phylogenetic Tree Shapes, Systematic Biology, volume 67, number 1, January 2018, pages 113126, with section 2.3 number of tips v_a(1) = a(n) for tree R_a = n+1.


FORMULA



EXAMPLE

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.


MATHEMATICA

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 *)


PROG



CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



