A043261 - OEIS

%I #24 Jun 14 2024 01:49:18

%S 0,1,2,2,3,2,4,2,3,4,5,2,4,4,6,2,3,4,5,4,5,6,7,2,4,4,6,4,6,6,8,2,3,4,

%T 5,4,5,6,7,4,5,6,7,6,7,8,9,2,4,4,6,4,6,6,8,4,6,6,8,6,8,8,10,2,3,4,5,4,

%U 5,6,7,4,5,6,7,6,7,8,9,4,5,6,7,6,7,8,9,6,7,8

%N Sum of the binary digits of the n-th base-2 palindrome.

%H Robert Israel, <a href="/A043261/b043261.txt">Table of n, a(n) for n = 1..10000</a>

%F Let b(1) = 0, b(2) = 1, otherwise b(2*n-1) = b(n-1) and b(2*n) = b(n).

%F Let c(1) = 0, c(2) = 1, otherwise c(2*n-1) = c(n-1)+1 and c(2*n) = c(n).

%F Then for n >= 2, a(2*n-1) = 2*c(2*n-1) - b(2*n-1) and a(2*n) = 2*c(2*n).

%e The fourth base-2 palindrome is 5 = 101_2, so a(4) = 1+0+1 = 2.

%p b:= proc(n) option remember;

%p   procname(floor(n/2)) end proc;

%p b(1):= 0; b(2):= 1;

%p c:= proc(n) option remember;

%p   procname(floor(n/2)) + (n mod 2) end proc;

%p c(1):= 0; c(2):= 1;

%p A043261:= n -> 2*c(n) - (n mod 2)*b(n);

%p A043261(2):= 1;# _Robert Israel_, Apr 06 2014

%o (Python)

%o def A043261(n):

%o     if n == 1: return 0

%o     a = 1<<(l:=n.bit_length()-2)

%o     m = a|(n&a-1)

%o     return (m.bit_count()<<1) - (0 if a&n else m&1) # _Chai Wah Wu_, Jun 13 2024

%Y Cf. A006995 (base-2 palindromes), A057148.

%K nonn,base

%O 1,3

%A _Clark Kimberling_

%E edited by _Robert Israel_, Apr 06 2014