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A230426 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of terminal nodes (leaves) in that finite branch of the beanstalk.

%I A230426 #9 Nov 21 2013 05:18:31
%S A230426 0,0,0,1,1,0,2,0,1,1,0,3,0,1,1,2,2,0,1,1,1,1,1,0,2,0,1,1,0,3,0,1,1,2,
%T A230426 4,0,1,1,2,2,0,1,1,1,1,1,0,3,0,1,1,2,0,3,1,1,3,0,2,1,1,2,3,0,1,1,1,1,
%U A230426 1,2,0,3,1,1,0,6,2,1,1,0,4,2,1,1,2,0,3
%N A230426 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of terminal nodes (leaves) in that finite branch of the beanstalk.
%C A230426 This sequence relates to the factorial base representation (A007623) in the same way as A213726 relates to the binary system.
%H A230426 Antti Karttunen, <a href="/A230426/b230426.txt">Table of n, a(n) for n = 0..10080</a>
%F A230426 If A230412(n)=0, a(n)=1; otherwise, if n is in A219666, a(n)=0; otherwise a(n) = a(A230423(n)) + a(A230424(n)).
%e A230426 From 11 sprouts the following finite side-tree of "factorial beanstalk":
%e A230426     18  19
%e A230426      \  /
%e A230426   14  15
%e A230426    \  /
%e A230426     11
%e A230426 Its leaves are the numbers 14, 18 and 19 (which all occur in A219658), whose factorial base representations (see A007623) are '210', '300' and '301' respectively. The corresponding parent nodes are obtained by subtracting the sum of factorial base digits, thus we get 18-3 = 15 and also 19-4 = 15, thus 15 ('211' in factorial base) is the parent of 18 and 19. For 14 and 15 we get 14-3 = 15-4 = 11, thus 11 is the parent of both 14 and 15, and the common ancestor of all numbers 11, 14, 15, 18 and 19.
%e A230426 For numbers not occurring in A219666 this sequence gives the number of leaves in such subtrees. Thus a(11)=3, a(14)=1 (counting just the leaf 14 itself), a(15)=2 and a(18) = a(19) = 1.
%o A230426 (Scheme with memoization-macro definec from Antti Karttunen's IntSeq-library)
%o A230426 (definec (A230426 n) (cond ((zero? (A230412 n)) 1) ((inA219666? n) 0) (else (+ (A230426 (A230423 n)) (A230426 (A230424 n))))))
%o A230426 (define (inA219666? n) (or (zero? n) (= 1 (- (A230418 (1+ n)) (A230418n)))))
%Y A230426 A219658 gives the position of ones in this sequence (which are the leaves of the tree).
%Y A230426 Differs from A230425 for the first time at n=34, where a(n)=4, while A230425(34)=3. Cf. also A230427.
%K A230426 nonn
%O A230426 0,7
%A A230426 _Antti Karttunen_, Nov 10 2013