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A255125 Number of times a multiple of four is encountered when iterating from 2^(n+1)-2 to (2^n)-2 with the map x -> x - (number of runs in binary representation of x). 9
1, 0, 1, 1, 1, 3, 6, 13, 26, 47, 81, 140, 253, 482, 949, 1875, 3666, 7088, 13614, 26100, 50082, 96246, 185131, 356123, 684758, 1316197, 2530257, 4868019, 9378335, 18096921, 34974646, 67669905, 130998912, 253565649, 490501587, 947992195, 1830664188, 3533571444 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Also the number of even numbers in range [A255062(n) .. A255061(n+1)] of A255057 (equally, in A255067). See the sum-formulas.

LINKS

Table of n, a(n) for n=0..37.

A. Karttunen, Ratio a(n)/A255126(n) plotted, with the help of OEIS-server

FORMULA

a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A059841(A255057(k)).

a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A059841(A255067(k)).

a(n) = A255071(n) - A255126(n).

EXAMPLE

For n=5 we start iterating with map m(n) = A236840(n) from the initial value (2^(5+1))-2 = 62. Thus we get m(62) = 60, m(60) = 58, m(58) = 54, m(54) = 50, m(50) = 46, m(46) = 42, m(42) = 36, m(36) = 32 and finally m(32) = 30, which is (2^5)-2. Of the nine numbers encountered, only 60, 36 and 32 are multiples of four, thus a(5) = 3.

PROG

(PARI)

A005811(n) = hammingweight(bitxor(n, n\2));

write_A255125_and_A255126_and_A255071(n) = { my(k, i, s25, s26); k = (2^(n+1))-2; i = 1; s25 = 0; s26 = 0; while(1, if((k%4), s26++, s25++); k = k - A005811(k); if(!bitand(k+1, k+2), break, i++)); write("b255125.txt", n, " ", s25); write("b255126.txt", n, " ", s26); write("b255071.txt", n, " ", i); };

for(n=1, 42, write_A255125_and_A255126_and_A255071(n));

(Scheme) (define (A255125 n) (if (zero? n) 1 (let loop ((i (- (expt 2 (+ 1 n)) 4)) (s 0)) (cond ((pow2? (+ 2 i)) s) (else (loop (- i (A005811 i)) (+ s (A133872 i))))))))

;; Alternatively:

(define (A255125 n) (add (COMPOSE A059841 A255057) (A255062 n) (A255061 (+ 1 n))))

(define (A255125 n) (add (COMPOSE A059841 A255067) (A255062 n) (A255061 (+ 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. A005811, A059841, A133872, A236840, A255056, A255057, A255061, A255062, A255067, A255071, A255126, A255332.

Similar sequences: A218542, A255063.

Sequence in context: A244704 A032198 A079941 * A267367 A265385 A019300

Adjacent sequences:  A255122 A255123 A255124 * A255126 A255127 A255128

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 18 2015

STATUS

approved

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Last modified July 24 03:01 EDT 2019. Contains 325290 sequences. (Running on oeis4.)