A218542 Number of times when an even number is encountered, when going from 2^(n+1)-1 to (2^n)-1 using the iterative process described in A071542.

1, 0, 1, 1, 2, 3, 8, 12, 23, 44, 86, 163, 308, 576, 1074, 1991, 3680, 6800, 12626, 23644, 44751, 85567, 164941, 319694, 621671, 1211197, 2362808, 4614173, 9018299, 17635055, 34486330, 67408501, 131642673, 256795173, 500346954, 973913365, 1894371802, 3683559071

Offset

0,5

Comments

Ratio a(n)/A213709(n) develops as: 1, 0, 0.5, 0.333..., 0.4, 0.333..., 0.471..., 0.400..., 0.426..., 0.449..., 0.480..., 0.494..., 0.502..., 0.501..., 0.497..., 0.489..., 0.479..., 0.469..., 0.461..., 0.455..., 0.453..., 0.454..., 0.458..., 0.464..., 0.469..., 0.475..., 0.480..., 0.484..., 0.488..., 0.492..., 0.496..., 0.499..., 0.502..., 0.503..., 0.505..., 0.505..., 0.505..., 0.505..., 0.505..., 0.504..., 0.504..., 0.503..., 0.503..., 0.502..., 0.502..., 0.502..., 0.503..., 0.503... (See further comments at A218543).

Links

Antti Karttunen, Table of n, a(n) for n = 0..47

Formula

a(n) = Sum_{i=A218600(n) .. (A218600(n+1)-1)} A213728(i).

a(n) = A213709(n) - A218543(n).

Example

(2^0)-1 (0) is reached from (2^1)-1 (1) with one step by subtracting A000120(1) from 1. Zero is an even number, so a(0)=1.
(2^1)-1 (1) is reached from (2^2)-1 (3) with one step by subtracting A000120(3) from 3. One is not an even number, so a(1)=0.
(2^2)-1 (3) is reached from (2^3)-1 (7) with two steps by first subtracting A000120(7) from 7 -> 4, and then subtracting A000120(4) from 4 -> 3. Four is an even number, but three is not, so a(2)=1.

Prog

(Scheme with memoizing definec-macro): (definec (A218542 n) (if (zero? n) 1 (let loop ((i (- (expt 2 (1+ n)) n 2)) (s 0)) (cond ((pow2? (1+ i)) (+ s (- 1 (modulo i 2)))) (else (loop (- i (A000120 i)) (+ s (- 1 (modulo i 2)))))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1))))) ;; A004198 is bitwise AND
;; Or with a summing-function add:
(define (A218542v2 n) (add A213728 (A218600 n) (-1+ (A218600 (1+ n)))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

Crossrefs

Cf. A219662 (analogous sequence for factorial number system).

Sequence in context: A350440 A115449 A303851 * A194452 A242516 A282281

Adjacent sequences: A218539 A218540 A218541 * A218543 A218544 A218545

Keyword

nonn

Author

Antti Karttunen, Nov 02 2012

Extensions

More terms from Antti Karttunen, Jun 05 2013

Status

approved