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A233905
a(2n) = a(n), a(2n+1) = a(n) + n, with a(0)=0.
3
0, 0, 0, 1, 0, 2, 1, 4, 0, 4, 2, 7, 1, 7, 4, 11, 0, 8, 4, 13, 2, 12, 7, 18, 1, 13, 7, 20, 4, 18, 11, 26, 0, 16, 8, 25, 4, 22, 13, 32, 2, 22, 12, 33, 7, 29, 18, 41, 1, 25, 13, 38, 7, 33, 20, 47, 4, 32, 18, 47, 11, 41, 26, 57, 0, 32, 16, 49, 8, 42, 25, 60, 4, 40, 22, 59, 13, 51, 32, 71, 2, 42, 22, 63, 12, 54
OFFSET
0,6
COMMENTS
For every one bit in the binary representation of n, add the number represented by the substring left of it.
FORMULA
a(n) = sum(k=0..floor(log(n)/log(2)), bittest(n,k) * floor(n/2^(k+1)) ) = sum(k=0..A000523(n), A030308(n,k+1) * floor(n/2^(k+1)) ), with bittest(n,k)=0 or 1 according to the k-th bit of n (the zeroth bit the least significant).
a(n) = A011371(n) - A233931(n).
EXAMPLE
27 is 11011 in binary, so we add 1, 110=6, and 1101=13, so a(27)=20.
PROG
(PARI) a(n)=if(n<1, 0, if(n%2, a(n\2)+n\2, a(n/2)))
(PARI) a(n)=sum(k=0, floor(log(n)/log(2)), bittest(n, k)*floor(n/2^(k+1)))
(Scheme, with memoizing definec-macro from Antti Karttunen's IntSeq-library)
(definec (A233905 n) (cond ((zero? n) n) ((even? n) (A233905 (/ n 2))) (else (+ (A233905 (/ (- n 1) 2)) (/ (- n 1) 2)))))
;; Antti Karttunen, Dec 21 2013
CROSSREFS
Sequence in context: A286238 A286237 A059781 * A285284 A288183 A324055
KEYWORD
nonn
AUTHOR
Ralf Stephan, Dec 17 2013
STATUS
approved