A106487 - OEIS

%I #7 Nov 10 2022 07:40:57

%S 1,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,1,2,

%T 2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,

%U 3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5

%N Number of leaves in combinatorial game trees.

%C See the comment at A106486.

%C After n=0 differs from A000120 for the first time at n=64, where A000120(64)=1, while a(64)=2.

%e 3 = 2^0 + 2^1 = 2^(2*0) + 2^((2*0)+1) encodes the CGT tree \/ which has two terminal nodes, thus a(3)=2.

%e 64 = 2^6 = 2^(2*3), i.e. it encodes the CGT tree

%e \/

%e .\

%e which also has two terminal (non-root) nodes, so a(64)=2.

%o (Scheme:) (define (A106487 n) (cond ((zero? n) 1) (else (apply + (map A106487 (map shr (on-bit-indices n))))))) (define (shr n) (if (odd? n) (/ (- n 1) 2) (/ n 2))) (define (on-bit-indices n) (let loop ((n n) (i 0) (c (list))) (cond ((zero? n) (reverse! c)) ((odd? n) (loop (/ (- n 1) 2) (1+ i) (cons i c))) (else (loop (/ n 2) (1+ i) c)))))

%Y Cf. A000120, A106486.

%K nonn

%O 0,4

%A _Antti Karttunen_, May 21 2005