A231902 -id:A231902 - cp's OEIS frontend

A037800 Number of occurrences of 01 in the binary expansion of n.

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

Offset: 0

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 up-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..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, A033264, A056974, A056975, A056976, A056977, A056978, A056979, A056980.

Cf. A030308, A231902.

a037800 = f 0 . a030308_row where
   f c [_]          = c
   f c (1 : 0 : bs) = f (c + 1) bs
   f c (_ : bs)     = f c bs
-- Reinhard Zumkeller, Feb 20 2014

Table[SequenceCount[IntegerDigits[n,2],{0,1}],{n,0,120}] (* Harvey P. Dale, Aug 10 2023 *)

a(n) = { if(n == 0, 0, -1 + hammingweight(bitnegimply(n, n>>1))) };  \\ Gheorghe Coserea, Aug 31 2015

a(2n) = a(n), a(2n+1) = a(n) + [n is even]. - Ralf Stephan, Aug 21 2003
G.f.: 1/(1-x) * Sum_{k>=0} t^5/(1+t)/(1+t^2) where t=x^2^k. - Ralf Stephan, Sep 10 2003
a(n) = A069010(n) - 1, n>0. - Ralf Stephan, Sep 10 2003
Sum_{n>=1} a(n)/(n*(n+1)) = log(2)/2 + Pi/4 - 1 = A231902 - 1 (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A196521 Decimal expansion of Pi/4-log(2)/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.438824573117475654907044785090787437011542282663648828183396143330257...

L. B. W. Jolley, Summation of series, Dover Publications Inc., New York, 1961, p. 14 (eq. 72).

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.

Cf. A003881, A016655 (10*log(2)/2), A033264.

Cf. A231902 (Pi/4+log(2)/2), A342316.

Digits:=100; evalf(Pi/4-log(2)/2); # Wesley Ivan Hurt, Dec 06 2013

RealDigits[Pi/4 - Log[2]/2, 10, 100] (* Wesley Ivan Hurt, Dec 06 2013 *)

Pi/4-log(2)/2 \\ Altug Alkan, Dec 14 2015

Equals 1 - 1/2 - 1/3 + 1/4 + 1/5 - ....
Equals Sum_{n>=0} 2/((4*n+2)*(4*n+3)). - Peter Luschny, Dec 06 2013
Equals Sum_{n>=1} (-1)^(n+1)/((2*n-1)*(2*n)). - Robert FERREOL, Dec 14 2015
Equals Integral_{x=0..1} (arctan(x)) dx = Integral_{x=0..Pi/4} (x / cos(x)^2) dx = Integral_{x=0..1/sqrt(2)} (arcsin(x)/(1-x^2)^(3/2)) dx. - Robert FERREOL, Dec 14 2015
Equals Integral_{x>=0} (exp(x) - 1)/(exp(2*x) + 1) dx. - Peter Bala, Nov 01 2019
From Bernard Schott, Sep 07 2020: (Start)
Equals Sum_{n>=1} (-1)^(n*(n-1)/2) / n [compare with A231902 formula].
Equals Sum_{n>=0} (8*n+5) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
Equals Sum_{k>=1} A033264(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
From Peter Bala, Mar 04 2025: (Start)
Equals (1/2) * A342316.
Equals Integral_{x = 0..1} x/(x^2 - 2*x + 2) = Integral_{x = 0..1} x*(1 + x)/(2 - x^2*(1 - x)) dx.
Equals (5/2)*Sum_{n >= 1} 1/(n*binomial(3*n, n)*2^n). The first 10 terms of the series gives the approximate value 0.43882457311(68...), correct to 11 decimal places. (End)

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

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A037800 Number of occurrences of 01 in the binary expansion of n.

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A196521 Decimal expansion of Pi/4-log(2)/2.

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

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