A125989 - OEIS

%I #3 Mar 31 2012 13:21:13

%S 0,0,1,0,2,0,1,0,3,1,2,-1,2,0,1,0,4,2,3,0,4,0,1,-2,3,1,2,-1,2,0,1,0,5,

%T 3,4,1,5,1,2,-1,6,2,3,-2,3,-1,0,-3,4,2,3,0,4,0,1,-2,3,1,2,-1,2,0,1,0,

%U 6,4,5,2,6,2,3,0,7,3,4,-1,4,0,1,-2,8,4,5,0,6,0,1,-4,5,1,2,-3,2,-2,-1

%N Sum of heights of 10's in binary expansion of n.

%C The 'height' of the digits in the binary expansion of n is here defined by the algorithm where, starting from the least significant bit and the height=0 and proceeding leftwards, all encountered 1-bits decrease the height by one and all 0-bits increase it by one. The sequence gives the sums of heights at the positions where 0 changes to 1 when scanning the binary expansion from right to left. This sequence is used for computing A126302.

%e E.g. the lattice path /\/\ is encoded by 10 as 1010 in binary and both peaks occur at height=1, thus a(10)=2.

%e In comparison, 11 is 1011 in binary, so the only peak '10' occurs at height -1:

%e .../

%e /\/

%e thus a(11)=-1.

%o (Scheme:) (define (A125989 n) (let loop ((n n) (s 0) (h 0)) (cond ((zero? n) s) ((= 2 (modulo n 4)) (loop (/ (- n 2) 4) (+ s h 1) h)) ((odd? n) (loop (/ (- n 1) 2) s (- h 1))) (else (loop (/ n 2) s (+ 1 h))))))

%Y A126302 = a(A014486(n)). Cf. A085198.

%K sign,base

%O 0,5

%A _Antti Karttunen_, Jan 02 2007