A025547 -id:A025547 - cp's OEIS frontend

A166107 A sequence related to the Madhava-Gregory-Leibniz formula for Pi.

2, -10, 46, -334, 982, -10942, 140986, -425730, 7201374, -137366646, 410787198, -9473047614, 236302407090, -710245778490, 20563663645710, -638377099140510, 1912749274005030, -67020067316087550, 2477305680740159850

Offset: 0

Johannes W. Meijer, Oct 06 2009, Feb 26 2013, Mar 02 2013

The EG1 matrix is defined in A162005. The first column of this matrix leads to the function PLS(z) = sum(2*eta(2*m-1)*z^(2*m-2), m=1..infinity) = 2*log(2) - Psi(z) - Psi(-z) + Psi(z/2) + Psi(-z/2). The values of this function for z=n+1/2 are related to Pi in a curious way.

Gauss's digamma theorem leads to PLS(z=n+1/2) = (-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1). Now we define PLS(z=n+1/2) = a(n)/p(n) with a(n) the sequence given above and for p(n) we choose the esthetically nice p(n) = (2*n-1)!!/(floor((n-2)/3)*2+1)!!, n=>0. For even values of n the limit(a(2*n)/p(2*n), n=infinity) = Pi and for odd values of n the limit(a(2*n+1)/p(2*n+1), n=infinity) = - Pi. We observe that the a(n)/p(n) formulas resemble the partial sums of the Madhava-Gregory-Leibniz series for Pi = 4*(1-1/3+1/5-1/7+ ...), see the examples. The 'extra term' that appears in the a(n)/p(n) formulas, i.e., 2/(2*n+1), speeds up the convergence of abs(a(n)/p(n)) significantly. The first appearance of a digit in the decimal expansion of Pi occurs here for n: 1, 3, 9, 30, 74, 261, 876, 3056, .., cf. A126809. [Comment modified by the author, Oct 09 2009]

			The first few values of a(n)/p(n) are: a(0)/p(0) = 2/1; a(1)/p(1) = - 4*(1) + 2/3 = -10/3; a(2)/p(2) = 4*(1-1/3) + 2/5 = 46/15; a(3)/p(3) = - 4*(1-1/3+1/5) + 2/7 = - 334/105; a(4)/p(4)= 4*(1-1/3+1/5-1/7) + 2/9 = 982/315; a(5)/p(5) = - 4*(1-1/3+1/5-1/7+1/9) + 2/11 = -10942/3465; a(6)/p(6) = 4*(1-1/3+1/5-1/7+1/9-1/11) + 2/13 = 140986/45045; a(7)/p(7) = - 4*(1-1/3+1/5-1/7+1/9-1/11+1/13) + 2/15 = - 425730/135135.

Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.
Peter Borwein, The amazing number Pi, Nieuw Archief voor de Wiskunde, September 2000, pp. 254-258.
Johannes W. Meijer, A modified Madhava-Gregory-Leibniz formula for Pi, Mar 02 2013.
Pacific Institute for the Mathematical Sciences, Pi in the Sky magazine, 2000-2013.
Wislawa Szymborska, The admirable number Pi, Miracle Fair, 2002.
Eric. W. Weisstein, Gauss's Digamma Theorem., from Wolfram MathWorld.
Wikipedia, Madhava of Sangamagrama.

Cf. A162005, A001147, A130823, A025547, A220747, A000796, A157142, A126809, A133766, A133767, A154633.

A166107 := n -> A220747 (n)*((-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1)): A130823 := n -> floor((n-1)/3)*2+1: A220747 := n -> doublefactorial(2*n+1) / doublefactorial(A130823(n)): seq(A166107(n), n=0..20);

a(n) = p(n)*(-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1) with
p(n) = doublefactorial(2*n+1)/doublefactorial(floor((n-1)/3)*2+1) = A220747(n)
PLS(z) = 2*log(2) - Psi(z) - Psi(-z) + Psi(z/2) + Psi(-z/2)
PLS(z=n+1/2) = a(n)/p(n) = (-1)^n*4*sum((-1)^(k+1)/(2*k-1), k=1..n) + 2/(2*n+1)
PLS(z=2*n+5/2) - PLS(z=2*n+1/2) = 2/(4*n+5) - 4/(4*n+3) + 2/(4*n+1) which leads to:
Pi = 2 + 16 * sum(1/((4*n+5)*(4*n+3)*(4*n+1)), n=0 .. infinity).
PLS (z=2*n +7/2) - PLS(z=2*n+3/2) = 2/(4*n+7) - 4/(4*n+5) + 2/(4*n+3) which leads to:
Pi = 10/3 - 16*sum(1/((4*n+7)*(4*n+5)*(4*n+3)), n=0 .. infinity).
The combination of these two formulas leads to:
Pi = 8/3 + 48* sum(1/((4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)), n=0 .. infinity).

A167588 The second column of the ED4 array A167584.

Original entry on oeis.org

1, 6, 41, 372, 4077, 53106, 795645, 13536360, 257055705, 5400196830, 124170067665, 3104906420700, 83818724048325, 2431059231544650, 75354930324303525, 2486926158748693200, 87036225272850632625, 3220532233879435917750, 125594424461427237941625

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..150

Equals the second column of the ED4 array A167584.
Other columns are A024199 and A167589.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor), A001147, A142970.

Table[(1/2)*(-1)^(n)*(2*n - 3)!!*((n) + (4*n^2 - 1)*Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}]), {n, 1, 50}] (* G. C. Greubel, Jun 17 2016 *)

a(n) = (1/2)*(-1)^(n)*(2*n-3)!!*(n+(4*n^2-1)*Sum_{k=0..n-1} ((-1)^(k+n)/(2*k+1))).
From Peter Bala, Nov 01 2016: (Start)
a(n) = (2*n + 1)!! * Sum_{k = 0..n-1} (-1)^(k-1)/((2*k - 1)*(2*k + 1)*(2*k + 3)).
a(n) ~ Pi * 2^(n-3/2) * ((n+1)/e)^(n+1).
E.g.f.: (4*x*sqrt(1 - 4*x^2) + 2*arcsin(2*x))/(8*(1 - 2*x)^(3/2)).
a(n) = 6*a(n-1) + (2*n - 5)*(2*n - 1)*a(n-2) with a(0) = 0, a(1) = 1.
The sequence b(n) := (2*n + 1)!! = (2*n + 2)!/((n + 1)!*2^(n+1)) satisfies the same recurrence with b(0) = 1 and b(1) = 3. This leads to the continued fraction representation a(n) = b(n)*[ 1/(3 - 3/(6 + 5/(6 + 21/(6 + ... + (2*n - 5)*(2*n - 1)/(6))))) ] for n >= 2.
As n -> infinity, a(n)/(A001147(n+1)) -> 1/2!*Pi/4 = 1/(3 - 3/(6 + 5/(6 + 21/(6 + ... + (2*n - 5)*(2*n - 1)/(6 + ...))))). Compare with the generalized continued fraction representation Pi = 3 + 1^2/(6 + 3^2/(6 + 5^2/(6 + ...))). See A142970. (End)

A167589 The third column of the ED4 array A167584.

Original entry on oeis.org

1, 10, 93, 1020, 13269, 198990, 3383145, 64276920, 1349846505, 31046064210, 776157686325, 20956154152500, 607730434609725, 18839602224969750, 621707822126431425, 21759750056864358000, 805111392478121276625

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..150

Equals the third column of the ED4 array A167584.
Other columns are A024199 and A167588.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor), A001147.

Table[(1/8)*(-1)^(n)*(2*n - 5)!!*((4*n^3 - 11*n) + (16*n^4 - 40*n^2 + 9)*(Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}])), {n, 1, 50}] (* G. C. Greubel, Jun 17 2016 *)

a(n) = (1/8)*(-1)^n*(2*n-5)!!*((4*n^3-11*n)+(16*n^4-40*n^2+9)*(Sum_{k=0..n-1} (-1)^(k+n)/(2*k+1) ) ).
From Peter Bala, Nov 01 2016: (Start)
a(n) = 3*(2*n + 3)!! * Sum_{k = 0..n-1} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)).
a(n) ~ Pi*2^(n - 5/2)*((n + 2)/e)^(n + 2).
E.g.f.: (6*arcsin(2*x) + 4*x*sqrt(1 - 4*x^2)*(5 - 8*x^2))/(32*(1 - 2*x)^(5/2)).
a(n) = 10*a(n) + (2*n - 7)*(2*n + 1)*a(n-2) with a(0) = 0, a(1) = 1.
The sequence b(n) := (2*n + 3)!! = (2*n + 4)!/((n + 2)!*2^(n+2)) = A001147(n+2) satisfies the same recurrence with b(0) = 3 and b(1) = 15. This leads to the continued fraction representation a(n) = 1/3*b(n)*( 1/(5 - 15/(10 - 7/(10 + 9/(10 + 33/(10 + ... + (2*n - 7)*(2*n + 1)/(10)))))) ) for n >= 2.
As n -> infinity, 3*a(n)/(A001147(n+2)) -> 9/4!*Pi/4 = 1/(5 - 15/(10 - 7/(10 + 9/(10 + 33/(10 + ... + (2*n - 7)*(2*n + 1)/(10 + ...)))))). (End)

A083390 m such that 2m + 1 divides lcm(1,3,5,...,2m - 1).

Original entry on oeis.org

7, 10, 16, 17, 19, 22, 25, 27, 28, 31, 32, 34, 37, 38, 42, 43, 45, 46, 47, 49, 52, 55, 57, 58, 59, 61, 64, 66, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 85, 87, 88, 91, 92, 93, 94, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 115, 117, 118, 122, 123, 124, 126

Offset: 1

Lekraj Beedassy, Jun 11 2003

nonn

Also m for which A025547(m)=A025547(m+1). Query: a(n) seems to be equal to A030343(n+4) - 1. Is this true?

While any odd number>1 can be the leg of a primitive Pythagorean triangle, the m-th odd number 2m+1=A061346 forms leg common to more than one PPT. - Lekraj Beedassy, Jul 12 2006

			10 is in the sequence because we have 2*10 - 1 = 19 and lcm(1,3,5,...,19)=166966608033225=7950790858725*21 which is divisible by 2*10 + 1 = 21.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A080765.

Select[Range[150],Divisible[LCM@@Range[1,2#-1,2],2#+1]&] (* Harvey P. Dale, Jan 22 2023 *)

isok(n) = {lc = 1; for (i = 1, 2*n-1, lc = lcm(lc, i);); return (lc % (2*n+1) == 0);} \\ Michel Marcus, Jul 27 2013

a(n) = (A061346(n)-1)/2. - David Wasserman, Oct 26 2004

More terms from David Wasserman, Oct 26 2004

A111877 a(n) = denominator of 3Sum_{j=0..n+1} 1/(2j+1).

Original entry on oeis.org

1, 5, 35, 105, 1155, 15015, 15015, 255255, 4849845, 4849845, 111546435, 557732175, 1673196525, 48522699225, 1504203675975, 1504203675975, 1504203675975, 55655536011075, 55655536011075, 2281876976454075, 98120709987525225

Offset: 0

Paul Barry, Aug 19 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001620, A025547, A350669 (numerators).

[Denominator((2*HarmonicNumber(2*n+4) - HarmonicNumber(n+2)))/3: n in [0..40]]; // G. C. Greubel, Jul 24 2023

f[x_]:= 2*x+1; a[1]= f[1]; a[n_]:= LCM[f[n], a[n-1]]; Array[a, 21]/3 (* Robert G. Wilson v, Jan 04 2013 *)

[denominator(2*harmonic_number(2*n+4,1) - harmonic_number(n+2,1))/3 for n in range(41)] # G. C. Greubel, Jul 24 2023

a(n) = denominator of (3/2)*(digamma(n+5/2) + 2*log(2) + euler_gamma).
a(n) = denominator of ( 3*Sum_{j=0..n+1} 1/(2*j+1) ).
a(n) = (1/3) * denominator of ( 2*H_{2*n+4} - H_{n+2} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023

Name edited by G. C. Greubel, Jul 24 2023

A217858 Odd part of lcm(1,2,3,...,n).

Original entry on oeis.org

1, 1, 3, 3, 15, 15, 105, 105, 315, 315, 3465, 3465, 45045, 45045, 45045, 45045, 765765, 765765, 14549535, 14549535, 14549535, 14549535, 334639305, 334639305, 1673196525, 1673196525, 5019589575, 5019589575, 145568097675, 145568097675, 4512611027925, 4512611027925, 4512611027925

Offset: 1

Joerg Arndt, Oct 13 2012

nonn

			a(8) is the least common multiple of {1,3,5,7}.

Cf. A003418, A025547, A128501, A216917.

A217858 := n -> ilcm(op(select(j->j mod 2 = 1, [$1..n]))):
seq(A217858(i), i = 1..33); # Peter Luschny, Oct 15 2012

a[n_] := LCM @@ Select[Range[n], OddQ]; Array[a, 33] (* Jean-François Alcover, Jun 27 2019 *)

A003418(n) = lcm(vector(n,k, k));
A000265(n) = n/2^valuation(n,2);
a(n) = A000265( A003418(n) );

def A217858(n): return lcm([j for j in (1..n) if is_odd(j)])
[A217858(n) for n in (1..33)]  # Peter Luschny, Oct 15 2012

a(n) = A000265( A003418(n) ).
a(n) = lcm_{1 <= k <= n, k odd}. - Peter Luschny, Oct 15 2012
a(n) = A025547(floor((n+1)/2)). - Georg Fischer, Nov 29 2022

A127676 Numerators of partial sums of a series for Pi*sqrt(2)/4.

Original entry on oeis.org

1, 4, 17, 104, 347, 4132, 50251, 47248, 848261, 16882724, 16189889, 357817912, 1856017421, 5753962988, 161845337077, 4871637351712, 5008383140437, 5137314884092, 185568039683479, 181286844605704, 7599727236867089

Offset: 0

Wolfdieter Lang, Mar 07 2007

Denominators coincide with A025547(n+1) for n=0..41, but then start to differ. See the W. Lang link. denominator(r(42))=7422822568422519986207785205976075 but the corresponding entry is A025547(43)=126187983663182839765532348501593275.

			Rationals r(n): [1, 4/3, 17/15, 104/105, 347/315, 4132/3465, ...].

E. Maor, Trigonometric Delights, Princeton University Press, 1998, p. 205.

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Ayoub, Euler and the Zeta Function, Am. Math. Monthly 81 (1974) 1067-1086, p. 1070.
W. Lang, Rationals and limit.

Cf. A025547 (denominators).

[Numerator((&+[(-1)^Floor(k/2)/(1+2*k): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 17 2018

Numerator[Table[Sum[(-1)^Floor[k/2]/(2*k + 1), {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Aug 17 2018 *)

a(n) = numerator(sum(k=0, n, (-1)^(k\2)/(2*k+1))); \\ Michel Marcus, Oct 03 2017

a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n) = Sum_{k=0..n} (-1)^floor(k/2)/(2*k+1).

A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725

Offset: 0

Wolfdieter Lang, Apr 06 2022

This is not the sequence A025547(n+1)_{n>=0}, because a(32) = 1420993851085122917681925 but A025547(33) = 18472920064106597929865025. Hence it is also not the sequence A350670.
The alternating sum Sum_{k=0..n} (-1)^k/(2*k+1) = (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2, with the Digamma function Psi(z).
Proof by subtracting twice the negative fractions from Sum_{k=0..n} 1/(2*k+1) = A350669(n)/A350670(n) (Abramowitz-Stegun, p. 258, eq. 6.3.4.), using Sum_{j=0..k} 1/(4*j + 3) = A074637((k+1)/4)/A074638(k+1) (Abramowitz-Stegun, p. 258, eqs. 6.3.6. with 6.3.5.) and, finally, replacing in the results for the even and odd n cases the formula for Psi(3/4) = -A200134.

Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.

Cf. A007509 (numerators).

Cf. A025547, A074637, A074638, A200134, A350669, A350670.

Denominator @ Accumulate @ Table[(-1)^k/(2*k + 1), {k, 0, 25}] (* Amiram Eldar, Apr 08 2022 *)

a(n) = denominator(sum(k=0, n, (-1)^k / (2*k+1))); \\ Michel Marcus, Apr 07 2022

from fractions import Fraction
def A352395(n): return sum(Fraction(-1 if k % 2 else 1,2*k+1) for k in range(n+1)).denominator # Chai Wah Wu, May 18 2022

a(n) = denominator( (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2), for n >= 0, with the Digamma function. See the above comment.

a(n) = denominator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - Vaclav Kotesovec, May 16 2022

A025548 a(n) = sum of the exponents in the prime factorization of the least common multiple of {1,3,5,...,2n-1}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 23, 24, 24, 24, 25, 26, 27, 27, 27, 28, 28, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 34, 34, 34, 35, 35, 36, 37, 37, 38, 38, 38, 39, 40, 40, 40, 40

Offset: 1

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A025547.

N:= 100: # for a(1) .. a(N)
V:= Vector(N):
p:= 2:
do
  p:= nextprime(p);
  if p > 2*N-1 then break fi;
  J:= [seq((p^i+1)/2, i = 1 .. ilog[p](2*N-1))];
  V[J]:= V[J] +~ 1;
od:
ListTools:-PartialSums(convert(V,list)); # Robert Israel, Dec 26 2024

a(n) = A001222(A025547(n)).

Corrected and extended by Ray Chandler, May 01 2007

A111878 a(n) = A111877(n+1)/5.

Original entry on oeis.org

1, 7, 21, 231, 3003, 3003, 51051, 969969, 969969, 22309287, 111546435, 334639305, 9704539845, 300840735195, 300840735195, 300840735195, 11131107202215, 11131107202215, 456375395290815, 19624141997505045, 19624141997505045

Offset: 0

Paul Barry, Aug 19 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A025547, A111877.

H:=HarmonicNumber; [Denominator((2*H(2*n+6) - H(n+3)))/15: n in [0..40]]; // G. C. Greubel, Jul 24 2023

With[{H=HarmonicNumber}, Table[Denominator[2*H[2*n+6] -H[n+3]]/15, {n, 0, 40}]] (* G. C. Greubel, Jul 24 2023 *)

h=harmonic_number; [denominator(2*h(2*n+6,1) - h(n+3,1))/15 for n in range(41)] # G. C. Greubel, Jul 24 2023

a(n) = (1/15)*denominator(digamma(n+7/2)/2 + log(2) + euler_gamma/2).
a(n) = denominator(f(n+2)/15), where f(n) = Sum_{j=0..n} 1/(2*j+1).
a(n) = (1/15) * denominator of ( 2*H_{2*n+6} - H_{n+3} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023

cp's OEIS Frontend

A166107 A sequence related to the Madhava-Gregory-Leibniz formula for Pi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A167588 The second column of the ED4 array A167584.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A167589 The third column of the ED4 array A167584.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A083390 m such that 2m + 1 divides lcm(1,3,5,...,2m - 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A111877 a(n) = denominator of 3*Sum_{j=0..n+1} 1/(2*j+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A217858 Odd part of lcm(1,2,3,...,n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A127676 Numerators of partial sums of a series for Pi*sqrt(2)/4.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A111877 a(n) = denominator of 3Sum_{j=0..n+1} 1/(2j+1).