A301336 -id:A301336

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A301896

a(n) = product of total number of 0's and total number of 1's in binary expansions of 0, ..., n.

+10
2

0, 1, 4, 8, 20, 35, 54, 72, 117, 165, 221, 280, 352, 425, 504, 576, 726, 875, 1036, 1200, 1386, 1575, 1776, 1976, 2214, 2451, 2700, 2944, 3216, 3479, 3750, 4000, 4455, 4897, 5355, 5808, 6300, 6789, 7296, 7800, 8364, 8925, 9504, 10080, 10695, 11305, 11931, 12544, 13260, 13965, 14688

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = A059015(n)*A000788(n).

a(2^k-1) = 2^(k-2)*(2^k*(k - 2) + 4)*k.

EXAMPLE

+---+-----+---+---+---+---+----------+

| n | bin.|0's|sum|1's|sum| a(n) |

+---+-----+---+---+---+---+----------+

| 0 | 0 | 1 | 1 | 0 | 0 | 1*0 = 0 |

| 1 | 1 | 0 | 1 | 1 | 1 | 1*1 = 1 |

| 2 | 10 | 1 | 2 | 1 | 2 | 2*2 = 4 |

| 3 | 11 | 0 | 2 | 2 | 4 | 2*4 = 8 |

| 4 | 100 | 2 | 4 | 1 | 5 | 4*5 = 20 |

| 5 | 101 | 1 | 5 | 2 | 7 | 5*7 = 35 |

| 6 | 110 | 1 | 6 | 2 | 9 | 6*9 = 54 |

+---+-----+---+---+---+---+----------+

bin. - n written in base 2;

0's - number of 0's in binary expansion of n;

1's - number of 1's in binary expansion of n;

sum - total number of 0's (or 1's) in binary expansions of 0, ..., n.

MAPLE

b:= proc(n) option remember; `if`(n=0, [1, 0], b(n-1)+

(l-> [add(1-i, i=l), add(i, i=l)])(Bits[Split](n)))

end:

a:= n-> (l-> l[1]*l[2])(b(n)):

seq(a(n), n=0..50); # Alois P. Heinz, Mar 01 2023

MATHEMATICA

Accumulate[DigitCount[Range[0, 50], 2, 0]] Accumulate[DigitCount[Range[0, 50], 2, 1]]

PROG

(Python)

def A301896(n): return (2+(n+1)*(m:=(n+1).bit_length())-(1<<m)-(k:=sum(i.bit_count() for i in range(1, n+1))))*k # Chai Wah Wu, Mar 01 2023

CROSSREFS

Cf. A000120, A000788, A023416, A059015, A071295, A301336.

KEYWORD

nonn,base

AUTHOR

Ilya Gutkovskiy, Mar 28 2018

STATUS

approved

A334841

a(0) = 0; for n > 0, a(n) = (number of 1's and 3's in base 4 representation of n) - (number of 0's and 2's in base 4 representation of n).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

Values are even for base 4 representations of n with an even number of digits, and odd for base 4 representations of n with an odd number of digits, except for a(0).

LINKS

Table of n, a(n) for n=0..88.

FORMULA

a(n) = 2*A139351(n) - A110592(n), n>0. - R. J. Mathar, Sep 02 2020

EXAMPLE

n in #odd #even

n base 4 digits - digits = a(n)

= ====== =======================

0 0 0 - 0

1 1 1 - 0 = 1

2 2 0 - 1 = -1

3 3 1 - 0 = 1

4 10 1 - 1 = 0

5 11 2 - 0 = 2

6 12 1 - 1 = 0

7 13 2 - 0 = 2

MAPLE

a:= n-> `if`(n=0, 0, add(`if`(i in [1, 3], 1, -1), i=convert(n, base, 4))):

seq(a(n), n=0..100); # Alois P. Heinz, May 30 2020

MATHEMATICA

a[0] = 0; a[n_] := Total[-(-1)^(r = Range[0, 3]) * DigitCount[n, 4, r]]; Array[a, 100, 0] (* Amiram Eldar, May 13 2020 *)

Join[{0}, Table[Total[If[EvenQ[#], -1, 1]&/@IntegerDigits[n, 4]], {n, 90}]] (* Harvey P. Dale, Sep 06 2020 *)

PROG

(R)

qnary = function(n, e, q){

e = floor(n/4)

q = n%%4

if(n == 0 ){return(0)}

if(e == 0){return(q)}

else{return(c(qnary(e), (q)))}

}

m = 400

s = seq(2, m)

v = c(0)

for(i in s){

x = qnary(i-1)

x[which(x%%2!=0)] = 1

x[which(x%%2==0)] = -1

v[i] = sum(x)

}

(Python)

import numpy as np

def qnary(n):

e = n//4

q = n%4

if n == 0 : return 0

if e == 0 : return q

if e != 0 : return np.append(qnary(e), q)

m = 400

v = [0]

for i in range(1, m+1) :

t = np.array(qnary(i))

t[t%2 != 0] = 1

t[t%2 == 0] = -1

v = np.append(v, np.sum(t))

(PARI) a(n) = my(ret=0); if(n, forstep(i=0, logint(n, 2), 2, if(bittest(n, i), ret++, ret--))); ret; \\ Kevin Ryde, May 24 2020

(Python)

def A334841(n):

return 2*bin(n)[-1:1:-2].count('1')-(len(bin(n))-1)//2 if n > 0 else 0 # Chai Wah Wu, Sep 03 2020

CROSSREFS

Cf. A053737, A010065, A037863, A007090, A301336, A333596.

KEYWORD

sign,easy,base

AUTHOR

Alexander Van Plantinga, May 13 2020

STATUS

approved

A333596

a(0) = 0; for n > 0, a(n) = a(n-1) + (number of 1's and 3's in base-4 representation of n) - (number of 0's and 2's in base-4 representation of n).

+10
1

0, 1, 0, 1, 1, 3, 3, 5, 3, 3, 1, 1, 1, 3, 3, 5, 4, 5, 4, 5, 6, 9, 10, 13, 12, 13, 12, 13, 14, 17, 18, 21, 18, 17, 14, 13, 12, 13, 12, 13, 10, 9, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 9, 10, 13, 12, 13, 12, 13, 14, 17, 18, 21, 19, 19, 17, 17, 17, 19, 19, 21, 19, 19

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

Local maxima values minus 1 are divisible by 4.

For a digit-wise recurrence, it's convenient to sum n terms so b(n) = a(n-1) = Sum_{i=0..n-1} A334841(i). Then b(4n+r) = 4*b(n) + r*A334841(n) + (1 if r even), for 0 <= r <= 3 and 4n+r >= 1. This is 4 copies of terms 0..n-1 and r copies of the following n. The new lowest digits cancel when r is odd, or net +1 when r is even. Repeatedly expanding gives the PARI code below. - Kevin Ryde, Jun 02 2020

LINKS

Table of n, a(n) for n=0..73.

EXAMPLE

n in #odd #even

n base 4 digits - digits + a(n-1) = a(n)

= ====== ===============================

0 0 0 - 0

1 1 1 - 0 + 0 = 1

2 2 0 - 1 + 1 = 0

3 3 1 - 0 + 0 = 1

4 10 1 - 1 + 1 = 1

5 11 2 - 0 + 1 = 3

6 12 1 - 1 + 3 = 3

7 13 2 - 0 + 3 = 5

MAPLE

a:= proc(n) option remember; `if`(n=0, 0, a(n-1) +add(

`if`(i in [1, 3], 1, -1), i=convert(n, base, 4)))

end:

seq(a(n), n=0..80); # Alois P. Heinz, May 30 2020

MATHEMATICA

f[n_] := Total[(-1)^(r = Range[0, 3]) * DigitCount[n, 4, r]]; a[0] = 0; a[n_] := a[n] = a[n - 1] - f[n]; Array[a, 100, 0] (* Amiram Eldar, Apr 24 2020 *)

PROG

(R)

qnary = function(n, e, q){

e = floor(n/4)

q = n%%4

if(n == 0 ){return(0)}

if(e == 0){return(q)}

else{return(c(qnary(e), (q)))}

}

m = 400

s = seq(2, m)

v = c(0)

for(i in s){

x = qnary(i-1)

x[which(x%%2!=0)] = 1

x[which(x%%2==0)] = -1

v[i] = sum(x, v[i-1])

}

(Python)

import numpy as np

def qnary(n):

e = n//4

q = n%4

if n == 0 : return 0

if e == 0 : return q

if e != 0 : return np.append(qnary(e), q)

m = 400

v = [0]

for i in range(1, m+1) :

t = np.array(qnary(i))

t[t%2 != 0] = 1

t[t%2 == 0] = -1

v = np.append(v, np.sum([np.sum(t), v[i-1]]))

(PARI) a(n) = my(v=digits(n+1, 4), s=0); for(i=1, #v, my(t=v[i]); v[i]=t*s+!(t%2); s-=(-1)^t); fromdigits(v, 4); \\ Kevin Ryde, May 30 2020

(PARI) b(n)=my(d=digits(n, 4)); -sum(i=1, #d, (-1)^d[i])

first(n)=my(s); concat(0, vector(n, k, s+=b(k))) \\ Charles R Greathouse IV, Jul 04 2020

(Python)

from itertools import accumulate

def A334841(n):

return 2*bin(n)[-1:1:-2].count('1')-(len(bin(n))-1)//2 if n > 0 else 0

A333596_list = list(accumulate(A334841(n) for n in range(10000))) # Chai Wah Wu, Sep 03 2020

CROSSREFS

Cf. A053737, A010065, A037863, A268289, A007090, A231664, A301336, A334841.

KEYWORD

nonn,base,easy

AUTHOR

Alexander Van Plantinga, Mar 27 2020

STATUS

approved

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