A161511 Number of 1...0 pairs in the binary representation of 2n.

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 6, 4, 5, 5, 6, 5, 7, 6, 7, 5, 8, 7, 8, 6, 9, 7, 8, 5, 6, 6, 7, 6, 8, 7, 8, 6, 9, 8, 9, 7, 10, 8, 9, 6, 10, 9, 10, 8, 11, 9, 10, 7, 12, 10, 11, 8, 12, 9, 10, 6, 7, 7, 8, 7, 9, 8, 9, 7, 10, 9, 10, 8, 11, 9, 10, 7, 11, 10, 11, 9, 12, 10, 11, 8, 13, 11, 12, 9, 13

Offset: 0

Author

Franklin T. Adams-Watters, Jun 11 2009

Keywords

• nonn
• look

Comments

Row (partition) sums of A125106.

a(n) is also the weight (= sum of parts) of the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

Examples

			For n = 5, the binary representation of 2n is 1010; the 1...0 pairs are 10xx, 1xx0, and xx10, so a(5) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Peter J. Taylor, Recursion for A161511, answer to question on MathOverflow (2025).

Crossrefs

Cf. A000120, A243499 (gives the corresponding products), A227183, A056239, A243503, A006068, A163511.

Sum of prime indices of A005940.

Row sums of A125106.

A reverse version is A359043, row sums of A242628.

A029837 adds up standard compositions, row sums of A066099.

A029931 adds up binary indices, row sums of A048793.

Cf. A059893, A228351, A230877, A231204, A253565, A358194, A359042, A359359.

Programs

Mathematica

a[0] = 0; a[n_] := If[EvenQ[n], a[n/2] + DigitCount[n/2, 2, 1], a[(n-1)/2] + 1]; Array[a, 93, 0] (* Jean-François Alcover, Sep 09 2017 *)

PARI

a(n)=local(t,k);t=0;k=1;while(n>0,if(n%2==0,k++,t+=k);n\=2);t

Python

def A161511(n):
    a, b = 0, 0
    for i, j in enumerate(bin(n)[:1:-1], 1):
        if int(j):
            a += i-b
            b += 1
    return a # Chai Wah Wu, Jul 26 2023

Scheme

;; Two variants, the recursive one requiring memoizing definec-macro from Antti Karttunen's IntSeq-library.
(define (A161511 n) (let loop ((n n) (i 1) (s 0)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ i 1) s)) (else (loop (/ (- n 1) 2) i (+ s i))))))
(definec (A161511 n) (cond ((zero? n) n) ((even? n) (+ (A000120 n) (A161511 (/ n 2)))) (else (+ 1 (A161511 (/ (- n 1) 2))))))
;; Antti Karttunen, Jun 28 2014

Formula

a(0) = 0; a(2n) = a(n) + A000120(n); a(2n+1) = a(n) + 1.

From Antti Karttunen, Jun 28 2014: (Start)

Can be also obtained by mapping with an appropriate permutation from the lists of partition sizes computed for other enumerations similar to A125106:

a(n) = A227183(A006068(n)).

a(n) = A056239(A005940(n+1)).

a(n) = A243503(A163511(n)). (End)

a(n) = A029931(n) - binomial(A000120(n),2). - Gus Wiseman, Jan 03 2023

a(n) = a(n - A048896(n-1)) + 1 for n>=1 (see Peter J. Taylor link). - Mikhail Kurkov, Jul 04 2025