A069010 Number of runs of 1's in the binary representation of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 3, 2, 3

Offset: 0

Author

Henry Bottomley, Apr 02 2002

Keywords

• base
• easy
• nonn

Comments

a(n) is also the number of distinct parts in 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

Positions of first occurrences of k are A002450(k). - John Keith, Aug 30 2021

Examples

			a(11) = 2 since 11 is 1011 in binary with two runs of 1's.
a(12) = 1 since 12 is 1100 in binary with one run of 1's.
		

Links

• Tilman Piesk (terms 0..9999) & Antti Karttunen, Table of n, a(n) for n = 0..16384
• Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 148.
• Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.
• Ralf Stephan, Some divide-and-conquer sequences ...
• Ralf Stephan, Table of generating functions

Crossrefs

Cf. A000035, A000120, A005811, A014081, A033264, A037800.

Cf. A268411 (parity of the terms), A268412 (positions of even terms), A268415 (of odd terms).

Cf. A002450 (positions of record highs).

Cf. also A227349, A246588.

Programs

Maple

f:= proc(n) option remember; if n::even then procname(n/2)
elif n mod 4 = 1 then 1 + procname((n-1)/2) else  procname((n-1)/2) fi end proc:
f(0):= 0:
map(f, [$0..1000]); # Robert Israel, Sep 06 2015

Mathematica

Count[Split@ IntegerDigits[#, 2], n_ /; First@ n == 1] & /@ Range[0, 120] (* Michael De Vlieger, Sep 05 2015 *)

PARI

a(n) = (1 + (hammingweight(bitxor(n, n>>1)))) >> 1;  \\ Gheorghe Coserea, Sep 05 2015

Python

def A069010(n):
    return sum(1 for d in bin(n)[2:].split('0') if len(d)) # Chai Wah Wu, Nov 04 2016

Scheme

(define (A069010 n) (/ (+ (A005811 n) (A000035 n)) 2)) ;; Antti Karttunen, Feb 05 2016

Formula

a(n) = ceiling(A005811(n)/2) = A005811(n) - A033264(n). If 2^k <= n < 3*2^(k-1) then a(n) = a(n-2^k)+1; if 3*2^(k-1) <= n < 2^(k+1) then a(n) = a(n-2^k).

a(2n) = a(n), a(2n+1) = a(n) + [n is even]. - Ralf Stephan, Aug 20 2003

G.f.: (1/(1-x)) * Sum_{k>=0} (t/(1+t))/(1+t^2), where t=x^2^k. - Ralf Stephan, Sep 07 2003

a(n) = A000120(n) - A014081(n) = A037800(n) + 1, n>0. - Ralf Stephan, Sep 10 2003