A227349 -id:A227349 - cp's OEIS frontend

A283972 a(n) = n minus (product of lengths of 1-runs in binary representation of n) = n - A227349(n).

0, 1, 1, 3, 4, 4, 4, 7, 8, 9, 9, 10, 11, 11, 11, 15, 16, 17, 17, 19, 20, 20, 20, 22, 23, 24, 23, 25, 26, 26, 26, 31, 32, 33, 33, 35, 36, 36, 36, 39, 40, 41, 41, 42, 43, 43, 43, 46, 47, 48, 47, 50, 51, 50, 49, 53, 54, 55, 53, 56, 57, 57, 57, 63, 64, 65, 65, 67, 68, 68, 68, 71, 72, 73, 73, 74, 75, 75, 75, 79, 80, 81, 81, 83, 84, 84, 84, 86, 87, 88, 87, 89

Offset: 1

Antti Karttunen, Apr 14 2017

Antti Karttunen, Table of n, a(n) for n = 1..10923
Index entries for sequences related to binary expansion of n

Cf. A227349.

Cf. A227190, A284560, A284581 for similar sequences.

Scheme
```
(define (A283972 n) (- n (A227349 n)))
```

a(n) = n - A227349(n).

A284562 Product / LCM of the lengths of 1-runs in binary representation of n: a(n) = A227349(n) / A284569(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 0

Antti Karttunen, Apr 14 2017

Antti Karttunen, Table of n, a(n) for n = 0..10922
Index entries for sequences related to binary expansion of n

Cf. A227349, A284569, A284558.

(define (A284562 n) (/ (A227349 n) (A284569 n)))

a(n) = A227349(n) / A284569(n).

A284582 a(n) = gcd(A227349(n), A227350(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Apr 14 2017

Antti Karttunen, Table of n, a(n) for n = 0..10922
Index entries for sequences related to binary expansion of n

Cf. A167489, A227349, A227350, A284583.

(define (A284582 n) (gcd (A227349 n) (A227350 n)))

a(n) = gcd(A227349(n), A227350(n)).

A167489(n) = a(n) * A284583(n).

A284583 a(n) = lcm(A227349(n), A227350(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 3, 6, 2, 2, 4, 6, 3, 4, 5, 5, 4, 3, 6, 4, 2, 2, 6, 3, 2, 1, 2, 2, 2, 3, 4, 4, 6, 2, 4, 2, 2, 4, 6, 3, 6, 3, 6, 4, 4, 5, 6, 6, 5, 4, 4, 6, 3, 6, 3, 6, 4, 2, 2, 4, 2, 6, 4, 4, 3, 2, 2, 2, 1, 2, 3, 6, 2, 2, 4, 6, 3, 4, 5, 10, 4, 6, 12, 4, 2, 4, 6, 6, 2, 2, 4, 4, 4, 6, 8, 12, 3, 6, 6, 6, 3, 6

Offset: 0

Antti Karttunen, Apr 14 2017

Cf. A167489, A227349, A227350, A284583.

(define (A284583 n) (lcm (A227349 n) (A227350 n)))

a(n) = lcm(A227349(n), A227350(n)).

A167489(n) = a(n) * A284582(n).

A253083 Partial sums of A227349.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 12, 13, 14, 15, 17, 19, 21, 24, 28, 29, 30, 31, 33, 34, 35, 37, 40, 42, 44, 46, 50, 53, 56, 60, 65, 66, 67, 68, 70, 71, 72, 74, 77, 78, 79, 80, 82, 84, 86, 89, 93, 95, 97, 99, 103, 105, 107, 111, 117, 120, 123, 126, 132, 136, 140, 145, 151, 152, 153, 154, 156, 157, 158

Offset: 0

N. J. A. Sloane, Feb 05 2015

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..8192
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Cf. A227349.

A278222 The least number with the same prime signature as A005940(n+1).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 6, 6, 12, 4, 12, 8, 16, 2, 6, 6, 12, 6, 30, 12, 24, 4, 12, 12, 36, 8, 24, 16, 32, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 4, 12, 12, 36, 12, 60, 36, 72, 8, 24, 24, 72, 16, 48, 32, 64, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 6, 30, 30, 60, 30, 210, 60, 120, 12, 60, 60, 180, 24, 120, 48, 96, 4, 12, 12

Offset: 0

Antti Karttunen, Nov 15 2016

This sequence can be used for filtering certain base-2 related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A005940(n+1)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Because the Doudna map n -> A005940(1+n) is an isomorphism from "unary-binary encoding of factorization" (see A156552) to the ordinary representation of the prime factorization of n, it follows that the equivalence classes of this sequence match with any such sequence b, where b(n) is computed from the lengths of 1-runs in the binary representation of n and the order of those 1-runs does not matter. Particularly, this holds for any sequence that is obtained as a "Run Length Transform", i.e., where b(n) = Product S(i), for some function S, where i runs through the lengths of runs of 1's in the binary expansion of n. See for example A227349.
However, this sequence itself is not a run length transform of any sequence (which can be seen for example from the fact that A046523 is not multiplicative).
Furthermore, this matches not only with sequences involving products of S(i), but with any sequence obtained with any commutative function applied cumulatively, like e.g., A000120 (binary weight, obtained in this case as Sum identity(i)), and A069010 (number of runs of 1's in binary representation of n, obtained as Sum signum(i)).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005940, A156552, A006068, A124859, A278159.
Similar sequences: A278217, A278219 (other base-2 related variants), A069877 (base-10 related), A278226 (primorial base), A278234-A278236 (factorial base), A278243 (Stern polynomials), A278233 (factorization in ring GF(2)[X]), A046523 (factorization in Z).
Cf. also A286622 (rgs-transform of this sequence) and A286162, A286252, A286163, A286240, A286242, A286379, A286464, A286374, A286375, A286376, A286243, A286553 (various other sequences involving this sequence).
Sequences that partition N into same or coarser equivalence classes: too many to list all here (over a hundred). At least every sequence listed under index-entry "Run Length Transforms" is included (e.g., A227349, A246660, A278159), and also sequences like A000120 and A069010, and their combinations like A136277.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Array[If[# == 1, 1, Times @@ MapIndexed[ Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]]] &@ f[# - 1, 1, 1] &, 99] (* Michael De Vlieger, May 09 2017 *)

A046523(n)=factorback(primes(#n=vecsort(factor(n)[, 2], , 4)), n)
a(n)=my(p=2, t=1); for(i=0,exponent(n), if(bittest(n,i), t*=p, p=nextprime(p+1))); A046523(t) \\ Charles R Greathouse IV, Nov 11 2021

from sympy import prime, factorint
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return a046523(a005940(n + 1)) # Indranil Ghosh, May 05 2017

(define (A278222 n) (A046523 (A005940 (+ 1 n))))

a(n) = A046523(A005940(1+n)).
a(n) = A124859(A278159(n)).
a(n) = A278219(A006068(n)).

Misleading part of the name removed by Antti Karttunen, Apr 07 2022

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

Original entry on oeis.org

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

Henry Bottomley, Apr 02 2002

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

			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.

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

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.

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

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

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

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

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

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

A106737 a(n) = Sum_{k=0..n} ({binomial(n+k,n-k)*binomial(n,k)} mod 2).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 4, 2, 4, 4, 6, 3, 6, 4, 5, 2, 4, 4, 6, 4, 8, 6, 8, 3, 6, 6, 9, 4, 8, 5, 6, 2, 4, 4, 6, 4, 8, 6, 8, 4, 8, 8, 12, 6, 12, 8, 10, 3, 6, 6, 9, 6, 12, 9, 12, 4, 8, 8, 12, 5, 10, 6, 7, 2, 4, 4, 6, 4, 8, 6, 8, 4, 8, 8, 12, 6, 12, 8, 10, 4, 8, 8, 12, 8, 16, 12, 16, 6, 12, 12, 18, 8, 16, 10, 12

Offset: 0

Benoit Cloitre, May 15 2005

nonn

The formula (the recurrence, if confirmed to be equal to sum binomial formula) implies that this is the run length transform of the sequence 1,2,3,4,5,... - N. J. A. Sloane, Feb 05 2015. Note: That sequence should be considered as a successor function a(n) = n+1, starting from offset 0. See also A020725. - Antti Karttunen, Oct 15 2016

The recurrence formula is correct. See paper in links. - Chai Wah Wu, Oct 16 2016

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Row sums of triangle in A253084.
Cf. A000005, A005940, A020725, A227349, A277335 (positions of odd terms).
Cf. also A153013.

Table[Sum[Mod[#, 2] &[Binomial[n + k, n - k] Binomial[n, k]], {k, 0, n}], {n, 0, 95}] (* Michael De Vlieger, Oct 17 2016 *)

a(n) = sum(k=0, n, (binomial(n+k,n-k)*binomial(n,k)) % 2); \\ Michel Marcus, Dec 08 2013

def A106737(n):
    return sum(int(not (~(n+k) & (n-k)) | (~n & k)) for k in range(n+1)) # Chai Wah Wu, Feb 09 2016
(Scheme, two mathematically equal implementations, based on RLT-interpretation)
;; The first one implements the given recurrence and uses memoization-macro definec:
(definec (A106737 n) (cond ((zero? n) 1) ((even? n) (A106737 (/ n 2))) ((= 1 (modulo n 4)) (* 2 (A106737 (/ (- n 1) 2)))) (else (- (* 2 (A106737 (/ (- n 1) 2))) (A106737 (/ (- n 3) 4))))))
;; This one applies the Run Length Transform explicitly to r -> r+1 function:
(define (A106737 n) (fold-left (lambda (a r) (* a (+ 1 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2))))) ;; See A227349 for the required other functions.
;; Antti Karttunen, Oct 15 2016

a(0)=1, a(2n) = a(n), a(4n+1) = 2*a(2n), a(4n+3) = 2*a(2n+1) - a(n).
From Antti Karttunen, Oct 15 2016: (Start)
a(n) = A000005(A005940(1+n)). [Follows from the Run Length Transform-interpretation.]
For n > 1, a(n^2) is always even. [Based on RLT-interpretation. n^2 = 1 modulo 4 for all odd n and ((2^k)*n)^2 = 2^(2k) * (n^2), thus the last 1-bit is always alone and contributes 2 to the product, making it even.]
(End)

A167489 Product of run lengths in binary representation of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 1, 2, 4, 2, 3, 4, 4, 3, 2, 4, 2, 1, 2, 3, 6, 4, 2, 4, 6, 3, 4, 5, 5, 4, 3, 6, 4, 2, 4, 6, 3, 2, 1, 2, 4, 2, 3, 4, 8, 6, 4, 8, 4, 2, 4, 6, 9, 6, 3, 6, 8, 4, 5, 6, 6, 5, 4, 8, 6, 3, 6, 9, 6, 4, 2, 4, 8, 4, 6, 8, 4, 3, 2, 4, 2, 1, 2, 3, 6, 4, 2, 4, 6, 3, 4, 5, 10, 8, 6, 12, 8, 4, 8

Offset: 0

Andrew Weimholt, Nov 05 2009

			a(56) = 9, because 56 in binary is written 111000 giving the run lengths 3,3 and 3x3 = 9.
a(99) = 12, because 99 in binary is written 1100011 giving the run lengths 2,3,2, and 2x3x2 = 12.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Row products of A101211 and A227736 (for n > 0).
Cf. A167490 (smallest number with binary run length product = n).
Cf. A167491 (members of A167490 sorted in ascending order).
Cf. A227355, A227184, A227190.
Cf. A227349, A227350, A227352, A246588, A284558, A284559, A284582, A284583.
Differs from similar A284579 for the first time at n=56, where a(56) = 9, while A284579(56) = 5.

import Data.List (group)
a167489 = product . map length . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013

Table[ Times @@ (Length /@ Split[IntegerDigits[n, 2]]), {n, 0, 100}](* Olivier Gérard, Jul 05 2013 *)

a(n) = {my(p=1, b=n%2, i=0); while(n!=0, n=n>>1; i=i+1; if((n%2)!=b, p=p*i; i=0; b=n%2)); p} \\ Indranil Ghosh, Apr 17 2017, after the Python Program by Antti Karttunen

def A167489(n):
  '''Product of run lengths in binary representation of n.'''
  p = 1
  b = n%2
  i = 0
  while (n != 0):
    n >>= 1
    i += 1
    if ((n%2) != b):
      p *= i
      i = 0
      b = n%2
  return(p)
# Antti Karttunen, Jul 24 2013 (Cf. Python program for A227184).

(define (A167489 n) (apply * (binexp->runcount1list n)))
(define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2)))))))
;; Antti Karttunen, Jul 05 2013

a(n) = A227349(n) * A227350(n) = A227355(A227352(2n+1)). - Antti Karttunen, Jul 25 2013

a(n) = A284558(n) * A284559(n) = A284582(n) * A284583(n). - Antti Karttunen, Apr 16 2017

A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jun 25 2003

nonn

The n-th runs of ones is given by: 3 - A003849(n) (infinite Fibonacci word) = A076662(n+1). Runs of zeros are given by: A085358 and are also directly related to the Fibonacci sequence. Coefficients of A(x)^3 are found in A085359.
a(n) = 0 iff some binary digit of n is 1 while the corresponding binary digit of 3*n is 0. - Robert Israel, Jul 12 2016
The Run Length Transform of [0,1,0,0,0,...], A063524, the characteristic function of 1. (See A227349 for the definition). - Antti Karttunen, Oct 15 2016

Robert Israel, Table of n, a(n) for n = 0..10000
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen, A. Petersen and G. Skordev, Automaticity of double sequences generated by one-dimensional linear cellular automata, Theoret. Comput. Sci. 186 (1997), 195-209.
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Discrete Appl. Math. 66 (1996), 1-22.
R. Bacher and C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238, DOI: 10.1007/978-3-642-23283-1_15.
Index entries for characteristic functions
Index entries for sequences computed with run length transform

Cf. A001764 (ternary trees), A085358 (runs of zeros), A076662 (runs of ones), A003849 (infinite Fibonacci word), A085359 (A(x)^3).
Cf. also A003714, A005940, A008966, A063524, A227349, A277010, A324964.
Absolute values of A132971.

[Binomial(3*n,n) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 09 2016

f:= proc(n) local L,Lp;
  L:= convert(n,base,2);
  Lp:= convert(3*n,base,2);
  if has(L-Lp[1..nops(L)],1) then 0 else 1 fi
end proc:
map(f, [$0..100]); # Robert Israel, Jul 12 2016

Table[Mod[Binomial[3 n, n], 2], {n, 0, 120}] (* Michael De Vlieger, Jul 08 2016 *)

A085357(n) = !bitand(n,n<<1); \\ Antti Karttunen, Aug 22 2019

def A085357(n): return int(not n&(n<<1)) # Chai Wah Wu, Jun 25 2025

G.f.: 1 + x*A(x)^3 = A(x) (Mod 2); a(n) = A001764(n) (Mod 2).
a(n) = binomial(3n, n) (mod 2). Characteristic function of Fibbinary numbers (i.e. a(n)=1 iff n is in A003714). - Benoit Cloitre, Nov 15 2003
Recurrence: a(0) = 1, a(2n) = a(4n+1) = a(n), a(4n+3) = 0.
a(n-2) = A000256(n)(mod 2), for n>2. - John M. Campbell, Jul 08 2016
a(n) = A000621(n+1)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A000625(n)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A008966(A005940(1+n)). [Follows from the Run Length Transform interpretation, see also A277010.] - Antti Karttunen, Oct 15 2016
a(n) = abs(A132971(n)) = abs(A008683(A005940(1+n))). - Antti Karttunen, May 30 2017

cp's OEIS Frontend

A283972 a(n) = n minus (product of lengths of 1-runs in binary representation of n) = n - A227349(n).

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A284562 Product / LCM of the lengths of 1-runs in binary representation of n: a(n) = A227349(n) / A284569(n).

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A284582 a(n) = gcd(A227349(n), A227350(n)).

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A284583 a(n) = lcm(A227349(n), A227350(n)).

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A253083 Partial sums of A227349.

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A278222 The least number with the same prime signature as A005940(n+1).

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

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Maple

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A106737 a(n) = Sum_{k=0..n} ({binomial(n+k,n-k)*binomial(n,k)} mod 2).

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