A262696 - OEIS

%I #8 Oct 04 2015 13:10:34

%S 0,2,0,1,1,1,0,1,1,13,1,13,0,1,1,11,1,11,0,1,1,10,0,10,1,1,1,10,1,9,0,

%T 8,1,1,0,8,1,1,6,7,1,1,0,1,1,6,0,6,5,1,1,6,1,5,0,1,1,5,0,3,4,3,0,1,1,

%U 3,1,1,1,2,0,1,4,1,1,1,7,1,0,1,1,7,1,6,4,1,1,6,1,1,0,5,1,1,0,4,4,4,1,1,1,1,0,1,3,4,0,4,1,1,1,3,1,1,0,1,1,1,0,1,3,0,4,1

%N a(n)=0 if n is in A259934, otherwise number of terminal nodes (including n itself if it is a leaf) in that finite subtree whose root is n and whose edge-relation is defined by A049820(child) = parent.

%H Antti Karttunen, <a href="/A262696/b262696.txt">Table of n, a(n) for n = 0..17724</a>

%F If A262693(n) = 1 [when n is in A259934],

%F   then a(n) = 0,

%F otherwise, if A060990(n) = 0 [when n is one of the leaves, A045765],

%F   then a(n) = 1,

%F otherwise:

%F   a(n) = Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * a(k).

%F (In the last clause [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = n, and 0 otherwise).

%F Other identities:

%F For any n in A262511 but not in A259934, a(n) = a(A082284(n)).

%e For n=1, its transitive closure (as defined by edge-relation A049820(child) = parent) is the union of {1} itself together with all its descendants: {1, 3, 4, 5, 7, 8}. We see that there are no other nodes in a subtree whose root is 1, because A049820(3) = 3 - d(3) = 1, A049820(4) = 1, A049820(5) = 3, A049820(7) = 5, A049820(8) = 4 and of these only 7 and 8 are terms of A045765. Thus a(1) = 2.

%e For n=9, its transitive closure is {9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79}, of which only thirteen members: {13, 19, 24, 33, 36, 37, 43, 55, 63, 64, 67, 75, 79} are leaves (in A045765), thus a(9) = 13.

%o (Scheme with memoization-macro definec)

%o (definec (A262696 n) (cond ((= 1 (A262693 n)) 0) ((= 0 (A060990 n)) 1) (else (let loop ((s 0) (k (A262686 n))) (cond ((<= k n) s) ((= n (A049820 k)) (loop (+ s (A262696 k)) (- k 1))) (else (loop s (- k 1))))))))

%Y Cf. A000005, A045765, A049820, A060990, A082284, A259934, A262686, A262693.

%Y Cf. A262522, A262695, A262697.

%Y Cf. also A213726.

%K nonn

%O 0,2

%A _Antti Karttunen_, Oct 04 2015