A196521 -id:A196521 - cp's OEIS frontend

A033264 Number of blocks of {1,0} in the binary expansion of n.

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

Offset: 1

Clark Kimberling

Number of i such that d(i) < d(i-1), where Sum_{d(i)*2^i: i=0,1,....,m} is base 2 representation of n.

This is the base-2 down-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Index entries for sequences related to binary expansion of n

Cf. A014081, A014082, A037800, A056974, A056975, A056976, A056977, A056978, A056979, A056980, A196521.
a(n) = A005811(n) - ceiling(A005811(n)/2) = A005811(n) - A069010(n).
Equals (A072219(n+1)-1)/2.
Cf. also A175047, A030308.
Essentially the same as A087116.

a033264 = f 0 . a030308_row where
   f c [] = c
   f c (0 : 1 : bs) = f (c + 1) bs
   f c (_ : bs) = f c bs
-- Reinhard Zumkeller, Feb 20 2014, Jun 17 2012

f:= proc(n) option remember; local k;
k:= n mod 4;
if k = 2 then procname((n-2)/4) + 1
elif k = 3 then procname((n-3)/4)
else procname((n-k)/2)
fi
end proc:
f(1):= 0: f(0):= q:
seq(f(i),i=1..100); # Robert Israel, Aug 31 2015

Table[Count[Partition[IntegerDigits[n, 2], 2, 1], {1, 0}], {n, 102}] (* Michael De Vlieger, Aug 31 2015, after Robert G. Wilson v at A014081 *)
Table[SequenceCount[IntegerDigits[n,2],{1,0}],{n,110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 26 2017 *)

a(n) = { hammingweight(bitand(n>>1, bitneg(n))) }; \\ Gheorghe Coserea, Aug 30 2015

def A033264(n): return ((n>>1)&~n).bit_count() # Chai Wah Wu, Jun 25 2025

G.f.: 1/(1-x) * Sum_(k>=0, t^2/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan, Sep 10 2003
a(n) = A069010(n) - (n mod 2). - Ralf Stephan, Sep 10 2003
a(4n) = a(4n+1) = a(2n), a(4n+2) = a(n)+1, a(4n+3) = a(n). - Ralf Stephan, Aug 20 2003
a(n) = A087116(n) for n > 0, since strings of 0's alternate with strings of 1's, which end in (1,0). - Jonathan Sondow, Jan 17 2016
Sum_{n>=1} a(n)/(n*(n+1)) = Pi/4 - log(2)/2 (A196521) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A231902 Decimal expansion of Pi/4 + log(2)/2.

Original entry on oeis.org

1, 1, 3, 1, 9, 7, 1, 7, 5, 3, 6, 7, 7, 4, 2, 0, 9, 6, 4, 3, 2, 4, 2, 7, 6, 9, 0, 6, 5, 4, 8, 9, 6, 4, 0, 0, 5, 0, 8, 7, 0, 4, 2, 4, 1, 7, 0, 2, 3, 9, 0, 4, 0, 8, 2, 3, 0, 4, 0, 7, 6, 1, 5, 2, 8, 2, 3, 6, 5, 0, 9, 1, 2, 5, 5, 6, 3, 9, 9, 6, 0, 7, 4, 5, 9, 9, 4

Offset: 1

Bruno Berselli, Nov 15 2013

			1.131971753677420964324276906548964005087042417023904082304076152823650...

L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 28 (formula 154).
Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.15, p. 269.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Michael Penn, A nice integral, YouTube video, 2022.

Cf. A003881 (Pi/4), A016655 (10*(log(2)/2)), A072691 (Pi^2/12).
Cf. A006752 (Catalan's constant)
Cf. A196521 (Pi/4-log(2)/2).
Cf. A037800.

SetDefaultRealField(RealField(100)); R:=RealField(); (Pi(R) + 2*Log(2))/4; // G. C. Greubel, Aug 24 2018

RealDigits[Pi/4 + Log[2]/2, 10, 90][[1]]

default(realprecision, 100); (Pi + 2*log(2))/4 \\ G. C. Greubel, Aug 24 2018

Equals 1 + Sum_{m>=1} -(-1)^m/(2*m*(2*m+1)) = 1 + 1/(2*3) - 1/(4*5) + 1/(6*7) - 1/(8*9) + ... .
From Amiram Eldar, Jul 16 2020: (Start)
Equals Integral_{x=1..oo} arctan(x)/x^2 dx.
Equals 1 + Integral_{x=0..1/2} log(4*x^2 + 1) dx. (End)
From Bernard Schott, Sep 07 2020: (Start)
Equals -Sum_{n>=1} (-1)^(n*(n+1)/2) / n [compare with A196521 formula].
Equals Sum_{n>=0} (32*n^2+40*n+11) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
Equals 1 + Sum_{k>=1} A037800(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A307485 A permutation of the nonnegative integers: one odd, two even, four odd, eight even, etc.; extended to nonnegative integer with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 7, 9, 6, 8, 10, 12, 14, 16, 18, 20, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 43, 45, 47, 49, 51, 53, 55

Offset: 0

M. F. Hasler, Apr 18 2019

The simple idea of "list the first odd number, first two even numbers, next four odd numbers, next eight even numbers..." leads to a permutation of the positive integers, which can quite naturally be extended to a permutation of the nonnegative integers, with a(0) = 0.

			The first odd number is a(1) = 1,
the first two even numbers are a(2..3) = (2, 4),
the next four odd numbers are a(4..7) = (3, 5, 7, 9),
the next eight even numbers are a(8..15) = (6, 8, ..., 20), etc.
the next sixteen odd numbers are a(16..31) = (11, 13, ..., 41),
the next thirty-two even numbers are a(32..63) = (22, 24, ..., 84), etc.
the next 64 odd numbers are a(64..127) = (43, 45, ..., 169),
the next 128 even numbers are a(128..255) = (86, 88, ..., 340), etc.

Orap Andrew a.k.a. Dalgerok, Codeforces Round #553 - C. Problem for Nazar, on Codeforces.com, April 2019.
Index entries for sequences that are permutations of the natural numbers.

Cf. A196521, A307613 (inverse permutation), A307612 (partial sums).
Cf. A103889 (odd & even swapped), A004442 (pairs reversed: n + (-1)^n).
Odd numbers: A005408. Even numbers: A005843.
Cf. A233275 (different permutation based on entangling odd & even numbers).

Join[{0},Flatten[Riffle[TakeList[Range[1,169,2],2^Range[0,6,2]],TakeList[Range[ 2,340,2],2^Range[ 1,7,2]]]]] (* Harvey P. Dale, Dec 17 2022 *)

PARI
```
A307485(n)=2*n-2^logint(n<<2+1,2)\3
```

Ignoring a(0) = 0, the k-th block (k >= 1) has 2^(k-1) terms, indexed from 2^(k-1) through 2^k-1, all having the same parity as k.
The difference between the last and the first term of this range is: a(2^k-1) - a(2^(k-1)) = 2^k - 2 = (2^(k-1) - 1)*2 = (starting index - 1) times two = ending index minus one.
The 1st, 3rd, ..., (2n+1)-th block = (n+1)-th odd block starts with A007583(n) = (1, 3, 11, 43, 171, ...), n >= 0.
The 2nd, 4th, ..., (2n+2)-th block = (n+1)-th even block starts with 2*A007583(n) = (2, 6, 22, 86, 342, ...), n >= 0, i.e., twice the starting value of the preceding odd block.
a(n) = 2*n - floor(2^k/3) where k = floor(log_2(4n+1)), n >= 0. (And 2^k == (-1)^k (mod 3) => floor(2^k/3) = (2^k-m)/3 with m = 1 if k even, m = 2 if k odd.)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 - log(2)/2 (A196521). - Amiram Eldar, Nov 28 2023

A374157 a(n) = (-1)^floor(n/2)*n.

Original entry on oeis.org

0, 1, -2, -3, 4, 5, -6, -7, 8, 9, -10, -11, 12, 13, -14, -15, 16, 17, -18, -19, 20, 21, -22, -23, 24, 25, -26, -27, 28, 29, -30, -31, 32, 33, -34, -35, 36, 37, -38, -39, 40, 41, -42, -43, 44, 45, -46, -47, 48, 49, -50, -51, 52, 53, -54, -55, 56, 57, -58, -59

Offset: 0

Peter Luschny, Jun 30 2024

For all odd numbers n (A005408) and all whole numbers z (A001057) K(z/n) = K(a(n)/z), where K(z/n) denotes the Kronecker symbol (A372728). This fact is equivalent to the law of quadratic reciprocity and its first and second supplement.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A005408, A001057, A057077, A196521, A372728.

Cf. A001477, A181983.

a := n -> (-1)^iquo(n, 2)*n: seq(a(n), n = 0..59);

Array[(-1)^Floor[#/2]*# &, 60, 0] (* Michael De Vlieger, Jun 30 2024 *)

a(n) = (-1)^(n\2) * n; \\ Amiram Eldar, Jun 30 2024

def A374157(n): return (-1)**(n // 2)*n

def A374157(n): return -n if n&2 else n # Chai Wah Wu, Jun 30 2024

Sum_{n>=1} 1/a(n) = Pi/4 - log(2)/2 = A196521.
a(n) = [x^n] -x*(x^2 + 2*x - 1)/(x^2 + 1)^2.
a(n) = n! * [x^n] x*(cos(x) - sin(x)). - Stefano Spezia, Jun 30 2024
a(n) = n*A057077(n). - Michel Marcus, Jul 01 2024

A174091 a(n) = n OR 2.

Original entry on oeis.org

2, 3, 2, 3, 6, 7, 6, 7, 10, 11, 10, 11, 14, 15, 14, 15, 18, 19, 18, 19, 22, 23, 22, 23, 26, 27, 26, 27, 30, 31, 30, 31, 34, 35, 34, 35, 38, 39, 38, 39, 42, 43, 42, 43, 46, 47, 46, 47, 50, 51, 50, 51, 54, 55, 54, 55, 58, 59, 58, 59, 62, 63, 62, 63, 66, 67, 66

Offset: 0

Gary Detlefs, Feb 06 2013

OR(n, 2) + AND(n, 2) = n + 2.
OR(n, 2) - AND(n, 2) = n + 2*(-1)^floor(n/2), A004443.
a(n) = n when n = 2 or 3 mod 4 (n is in A042964). - Alonso del Arte, Feb 07 2013

			a(3) = 3 because OR(0011, 0010) = 0011 = 3.
a(4) = 6 because OR(0100, 0010) = 0110 = 6.
a(5) = 7 because OR(0101, 0010) = 0111 = 7.

Shane Chern, T. Cai, and H. Zhong, On the cardinality and sum of reciprocals of primitive sequences, Preprint 2018; To appear in Adv. Math. (China).
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. similar sequences listed in A244587.

Cf. A004443, A042964, A196521.

Maple
```
with(Bits): seq(Or(n,2), n=0..60);
```

Table[BitOr[n, 2], {n, 0, 100}] (* Alonso del Arte, Feb 06 2013 *)
LinearRecurrence[{2,-2,2,-1},{2,3,2,3},80] (* Harvey P. Dale, Oct 25 2016 *)

a(n)=bitor(n,2) \\ Charles R Greathouse IV, Feb 27 2013

a(n) = n + 1 + (-1)^floor(n/2).
G.f.: ( 2-x+x^3 ) / ( (1+x^2)*(x-1)^2 ). - R. J. Mathar, Feb 27 2013
Sum_{n>=0} (-1)^n/a(n) = Pi/4 - log(2)/2 = A196521. - Peter McNair, Aug 05 2023

A339799 Decimal expansion of Sum_{m>=1} (-1)^floor(sqrt(m)) / m.

Original entry on oeis.org

1, 2, 9, 4, 0, 8, 1, 2, 2, 1, 8, 8, 3, 0, 9, 1, 0, 7, 6, 3, 0, 3, 8, 2, 1, 7, 1, 8, 3, 5, 6, 7, 3, 1, 2, 5, 0, 5, 0, 1, 1, 2, 2, 5, 9, 5, 3, 9, 9, 2, 0, 4, 3, 0, 2, 2, 7, 6, 5, 9, 2, 3, 3, 9, 5, 2, 7, 5, 5, 1, 7, 1, 2, 7, 9, 3, 8, 5, 1, 5, 7, 1, 2, 0, 9, 0, 3, 6, 2, 6, 1, 8, 4, 8, 6, 1, 4, 2, 7, 8, 9, 6, 0, 8, 2

Offset: 1

Bernard Schott, Dec 17 2020

When grouped by negative and positive packs = - (1+1/2+1/3) + (1/4+1/5+1/6+1/7+1/8) - (1/9+...+1/15) + (1/16+...+1/24) +...+ (-1)^k (1/k^2 +...+ 1/((k+1)^2-1)) + ...

Sum_{m>=1} (-1)^floor(sqrt(m)) / m^q is convergent iff q > 1/2.

			-1.2940812218830910763038217183567312505011225953992043022765923395275517127938...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.35, p. 287.
E. Ramis , C. Deschamps, J. Odoux, Analyse 2, Exercices avec solutions, Classes Préparatoires aux Grandes Ecoles Scientifiques, Masson, Paris, 1985, Exercice 1. 1.14, pp. 12-13.

Wikipedia, Digamma function.

Cf. A196521, A228639, A231902.

evalf(Sum((-1)^n*(Psi(n^2 + 2*n + 1) - Psi(n^2)), n = 1 .. infinity), 120); # Vaclav Kotesovec, Dec 18 2020

sumalt(k=1, (-1)^k * (psi(1 + 2*k + k^2) - psi(k^2))) \\ Vaclav Kotesovec, Dec 18 2020

Equals Sum_{m>=1} (-1)^floor(sqrt(m)) / m.
Equals Sum_{m>=1} (-1)^m * Sum_{k=m^2..(m+1)^2-1} 1/k.
Equals Sum_{m>=1} (-1)^m * (digamma((m+1)^2) - digamma(m^2)).

More terms from Vaclav Kotesovec, Dec 18 2020

A342316 Decimal expansion of Pi/2 - log(2).

Original entry on oeis.org

8, 7, 7, 6, 4, 9, 1, 4, 6, 2, 3, 4, 9, 5, 1, 3, 0, 9, 8, 1, 4, 0, 8, 9, 5, 7, 0, 1, 8, 1, 5, 7, 4, 8, 7, 4, 0, 2, 3, 0, 8, 4, 5, 6, 5, 3, 2, 7, 2, 9, 7, 6, 5, 6, 3, 6, 6, 7, 9, 2, 2, 8, 6, 6, 6, 0, 5, 1, 4, 5, 8, 1, 1, 7, 3, 4, 0, 9, 7, 8, 3, 7, 0, 8, 1, 5, 4, 0, 8, 5, 6, 7, 4, 6, 3, 9, 8, 4, 6, 4, 4, 9

Offset: 1

Peter Luschny, Mar 14 2021

			0.87764914623495130981408957018157487402308456532730...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 2.5.2.n, pp. 186 and 223.

Cf. A019669 (Pi/2), A002162 (log(2)), A196521.

Mathematica
```
RealDigits[N[Pi/2 - Log[2], 105]][[10]]
```

Pi/2 - log(2) \\ Michel Marcus, Mar 14 2021

Equals (-log(4) - psi(1/4) + psi(3/4)) / 2, where psi(x) denotes the digamma function.
Equals -Integral_{x=0..1} log(x)/((1+x)*sqrt(1-x^2)) dx. - Bernard Schott, Apr 28 2021
Equals Sum_{k>=1} (-1)^(k+1)/(k*(2*k-1)). - Amiram Eldar, Jun 08 2021
From Peter Bala, Mar 05 2024: (Start)
Equals  2 * A196521.
Equals (10/3)*Integral_{x = 0..1} x/(2 - x^2*(1 - x)) dx.
Equals 5*Sum_{n >= 1} 1/(n*binomial(3*n,n)*2^n). The first 10 terms of the series gives the approximate value 0.87764914623(37...), correct to 11 decimal places. (End)

cp's OEIS Frontend

A033264 Number of blocks of {1,0} in the binary expansion of n.

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A231902 Decimal expansion of Pi/4 + log(2)/2.

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A307485 A permutation of the nonnegative integers: one odd, two even, four odd, eight even, etc.; extended to nonnegative integer with a(0) = 0.

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A374157 a(n) = (-1)^floor(n/2)*n.

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A174091 a(n) = n OR 2.

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A339799 Decimal expansion of Sum_{m>=1} (-1)^floor(sqrt(m)) / m.

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