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a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the right side" construction.
12

%I #12 Nov 12 2012 17:59:54

%S 0,0,0,0,1,0,0,1,3,0,0,0,1,3,0,4,0,0,1,0,0,1,3,0,4,0,0,1,4,0,0,2,0,3,

%T 1,0,0,0,1,3,0,4,0,0,1,4,0,0,2,0,3,1,0,4,0,0,2,0,37,0,0,110,0,3,0,1,0,

%U 0,1,3,0,4,0,0,1,4,0,0,2,0,3,1,0,4,0,0

%N a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the right side" construction.

%C "Tendrils" of the beanstalk are the finite side-trees sprouting from its infinite trunk (see A179016) at the numbers given by A213730.

%H A. Karttunen, <a href="/A218788/b218788.txt">Table of n, a(n) for n = 1..8727</a>

%H A. Karttunen, <a href="/A014486/a014486_1.pdf">Illustration of how binary trees (the second rightmost column) are encoded by A014486</a>

%e A213730(9)=22, and from that branches 24 and 25 (as both A011371(24)=A011371(25)=22) and while 24 is a leaf (in A055938) the other branch 25 further branches to two leaves (as both A011371(28)=A011371(29)=25).

%e When we construct a binary tree from this in such a fashion that the larger numbers go to the left, we obtain:

%e ..........

%e 29...28...

%e ..\./.....

%e ...25..24.

%e ....\./...

%e .....22...

%e ..........

%e and the binary tree

%e .......

%e .\./...

%e ..*....

%e ...\./.

%e ....*..

%e .......

%e is located as A014486(3) in the normal encoding order of binary trees, thus a(9)=3.

%o (Scheme with Antti Karttunen's memoization macro definec):

%o (define (A218788 n) (Aux_for218788 (A213730 n)))

%o (definec (Aux_for218788 n) (cond ((zero? (A079559 n)) 0) ((not (zero? (A213719 n))) -1) (else (A072764bi (Aux_for218788 (A213724 n)) (Aux_for218788 (A213723 n))))))

%Y These are the mirror-images of binary trees given in A218787, i.e. a(n) = A057163(A218787(n)). A218786 gives the sizes of these trees. Cf. A072764, A218610, A218611.

%K nonn

%O 1,9

%A _Antti Karttunen_, Nov 11 2012