A123092 -id:A123092 - cp's OEIS frontend

A222068 Decimal expansion of (1/16)*Pi^2.

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

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in four dimensions (achieved for example by the D_4 lattice).
From Hugo Pfoertner, Aug 29 2018: (Start)
Also decimal expansion of Sum_{k>=0} (-1)^k*d(2*k+1)/(2*k+1), where d(n) is the number of divisors of n A000005(n).
Ramanujan's question 770 in the Journal of the Indian Mathematical Society (VIII, 120) asked "If d(n) denotes the number of divisors of n, show that d(1) - d(3)/3 + d(5)/5 - d(7)/7 + d(9)/9 - ... is a convergent series ...".
A summation of the first 2*10^9 terms performed by Hans Havermann yields 0.6168503077..., which is close to (Pi/4)^2=0.616850275...
(End)
From Robert Israel, Aug 31 2018: (Start)
Modulo questions about rearrangement of conditionally convergent series, which I expect a more careful treatment would handle, Sum_{k>=0} (-1)^k*d(2*k+1)/(2*k+1) should indeed be Pi^2/16.
Sum_{k>=0} (-1)^k d(2k+1)/(2k+1)
  = Sum_{k>=0} Sum_{2i+1 | 2k+1} (-1)^k/(2k+1)
(letting 2k+1=(2i+1)(2j+1): note that k == i+j (mod 2))
  = Sum_{i>=0} Sum_{j>=0} (-1)^(i+j)/((2i+1)(2j+1))
  = (Sum_{i>=0} (-1)^i/(2i+1))^2 = (Pi/4)^2. (End)
Volume bounded by the surface (x+y+z)^2-2(x^2+y^2+z^2)=4xyz, the ellipson (see Wildberger, p. 287). - Patrick D McLean, Dec 03 2020

			0.6168502750680849136771556874922594459571...

S. D. Chowla, Solution and Remarks on Question 770, J. Indian Math. Soc. 17 (1927-28), 166-171.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 507.
S. Ramanujan, Coll. Papers, Chelsea, 1962, Question 770, page 333.
G. N. Watson, Solution to Question 770, J. Indian Math. Soc. 18 (1929-30), 294-298.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
B. C. Berndt, Y. S. Choi and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q770, JIMS VIII).
J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
Mathematics StackExchange, Sum_k (-1)^k tau(2k+1)/(2k+1).
G. Nebe and N. J. A. Sloane, Home page for D_4 lattice.
N. J. A. Sloane and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
N. J. Wildberger, Divine Proportions: Rational Trigonometry to Universal Geometry, Wild Egg Books, Sydney 2005.
Index entries for transcendental numbers.

Cf. A000005.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222069, A222070, A222071, A222072, A260646.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor((1/16)*10^100*pi^2))); // Vincenzo Librandi, Feb 20 2017

RealDigits[N[Gamma[3/2]^4, 104]] (* Fred Daniel Kline, Feb 19 2017 *)
RealDigits[N[Pi^2/16, 100]][[1]] (* Vincenzo Librandi, Feb 20 2017 *)
Integrate[Boole[(x+y+z)^2-2(x^2+y^2+z^2)>4x y z],{x,0,1},{y,0,1},{z,0,1}] (* Patrick D McLean, Dec 03 2020 *)

(Pi/4)^2 \\ Charles R Greathouse IV, Oct 31 2014

Equals A003881^2. - Bruno Berselli, Feb 11 2013
Equals A123092+1/2. - R. J. Mathar, Feb 15 2013
Equals Integral_{x>0} x^2*log(x)/((1+x)^2*(1+x^2)) dx. - Jean-François Alcover, Apr 29 2013
Equals the Bessel moment integral_{x>0} x*I_0(x)*K_0(x)^3. - Jean-François Alcover, Jun 05 2016
Equals Sum_{k>=1} zeta(2*k)*k/4^k. - Amiram Eldar, May 29 2021

A069075 a(n) = (4*n^2 - 1)^2.

Original entry on oeis.org

1, 9, 225, 1225, 3969, 9801, 20449, 38025, 65025, 104329, 159201, 233289, 330625, 455625, 613089, 808201, 1046529, 1334025, 1677025, 2082249, 2556801, 3108169, 3744225, 4473225, 5303809, 6245001, 7306209, 8497225, 9828225, 11309769

Offset: 0

Benoit Cloitre, Apr 05 2002

Products of squares of 2 successive odd numbers. - Peter Munn, Nov 17 2019

L. B. W. Jolley, Summation of Series, Dover, 1961.
Konrad Knopp, Theory and application of infinite series, Dover, 1990, p. 269.

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000466, A123092, A166329.

(4*Range[0,30]^2-1)^2 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,9,225,1225,3969},30] (* Harvey P. Dale, Feb 23 2018 *)

Sum_{n>=1} 1/a(n) = (Pi^2 - 8)/16 = 0.1168502750680... (A123092) [Jolley eq. 247]
G.f.: (-1 - 4*x - 190*x^2 - 180*x^3 - 9*x^4) / (x-1)^5. - R. J. Mathar, Oct 03 2011
a(n) = A000466(n)^2. - Peter Munn, Nov 17 2019
E.g.f.: exp(x)*(1 + 8*x + 104*x^2 + 96*x^3 + 16*x^4). - Stefano Spezia, Nov 17 2019
Sum_{n>=0} (-1)^n/a(n) = Pi/8 + 1/2. - Amiram Eldar, Feb 08 2022

A248895 Decimal expansion of Sum_{i >= 1} 1/(4*i^2-1)^3.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Mar 06 2015

			0.0373622936989363147421332343808054155321703402855879393868742479896853989...

Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).

Cf. A123092: Sum_{i >= 1} 1/(4*i^2-1)^2.

Cf. A248896: Sum_{i >= 1} 1/(4*i^2-1)^4.

Join[{0}, RealDigits[1/2 - 3 (Pi/8)^2, 10, 100][[1]]]

Equals 1/2 - 3*(Pi/8)^2.

A248896 Decimal expansion of Sum_{i >= 1} 1/(4*i^2-1)^4.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Mar 06 2015

			0.0123661758680770778665039878710802673296075102733191933383702352559...

Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).

Cf. A123092: Sum_{i >= 1} 1/(4*i^2-1)^2.

Cf. A248895: Sum_{i >= 1} 1/(4*i^2-1)^3.

Join[{0}, RealDigits[(Pi^4 + 30 Pi^2 - 384)/768, 10, 100][[1]]]

Equals (Pi^4 + 30*Pi^2 - 384)/768.

A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n).

Original entry on oeis.org

1, -8, 1, 32, -3, -384, 30, 1, 1536, -105, -5, -30720, 1890, 105, 2, 61440, -3465, -210, -7, -10321920, 540540, 34650, 1512, 17, 4587520, -225225, -15015, -770, -17, -1486356480, 68918850, 4729725, 270270, 8415, 62, 2972712960, -130945815, -9189180, -567567, -21879, -341

Offset: 1

Sean A. Irvine, Apr 04 2025

The expansion of S(n) = Sum_{k>=1} (1 / (4*k^2-1)^n) in even powers of Pi was apparently first found by Euler and the solution for n<=4 appear in many tables of sums.

These sums have a natural denominator of 2^(2*n)*(n-1)! (or, more precisely, 2^(2*n+floor((n-1)/2))*(n-1)!), but sometimes (e.g., n=7, n=9) there are additional common factors leading to the "reduced" triangle presented here.

			S(1) = (        1                                                 ) / (2),
S(2) = (       -8 +        Pi^2                                   ) / (2^4) = A123092,
S(3) = (       32 -      3*Pi^2                                   ) / (2^5 * 2!) = A248895,
S(4) = (     -384 +     30*Pi^2 +       Pi^4                      ) / (2^7 * 3!) = A248896,
S(5) = (     1536 -    105*Pi^2 -     5*Pi^4                      ) / (2^7 * 4!),
S(6) = (   -30720 +   1890*Pi^2 +   105*Pi^4 +    2*Pi^6          ) / (2^9 * 5!),
S(7) = (    61440 -   3465*Pi^2 -   210*Pi^4 -    7*Pi^6          ) / (2^10 * 5!),
S(8) = (-10321920 + 540540*Pi^2 + 34650*Pi^4 + 1512*Pi^6 + 17*Pi^8) / (2^12 * 7!),
S(9) = (  4587520 - 225225*Pi^2 - 15015*Pi^4 -  770*Pi^6 - 17*Pi^8) / (2^18 * 5 * 7), ...

E. P. Adams, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, 1922 (eq. 6.911).

I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.235).
Sean A. Irvine, Computing A382782, 2025.
Sean A. Irvine, Java program (github)
L. B. W. Jolley, Summation of Series, Dover, (1961) (eq. 373).

Cf. A123092 (n=2), A248895 (n=3), A248896 (n=4).

Cf. A382783, A382784.

A382784 Irregular triangle T(n,k) read by rows of the coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n) with denominator 2^(2n)*(n-1)!.

Original entry on oeis.org

2, -8, 1, 64, -6, -768, 60, 2, 12288, -840, -40, -245760, 15120, 840, 16, 5898240, -332640, -20160, -672, -165150720, 8648640, 554400, 24192, 272, 5284823040, -259459200, -17297280, -887040, -19584, -190253629440, 8821612800, 605404800, 34594560, 1077120, 7936, 7610145177600, -335221286400, -23524300800, -1452971520, -56010240, -872960

Offset: 1

Sean A. Irvine, Apr 04 2025

sign

See A382782 for a version of this triangle where common factors have been removed.

			Triangle begins:
S(1) =  (2) / (2^2 * 0!),
S(2) = -(8 - Pi^2) / (2^4 * 1!) = A123092,
S(3) =  (64 - 6*Pi^2) / (2^6 * 2!) = A248895,
S(4) = -(768 - 60*Pi^2 - 2*Pi^4)/ (2^8 * 3!) = A248896,
S(5) =  (12288 - 840*Pi^2 - 40*Pi^4) / (2^10 * 4!),
S(6) = -(245760 - 15120*Pi^2 - 840*Pi^4 - 16*Pi^6) / (2^12 * 5!),
S(7) =  (5898240 - 332640*Pi^2 - 20160*Pi^4 - 672*Pi^6) / (2^14 * 6!),
S(8) = -(165150720 - 8648640*Pi^2 - 554400*Pi^4 - 24192*Pi^6 - 272*Pi^8) / (2^16 * 7!),
S(9) =  (5284823040 - 259459200*Pi^2 - 17297280*Pi^4 - 887040*Pi^6 - 19584*Pi^8) / (2^18 * 8!), ...

Cf. A123092 (n=2), A248895 (n=3), A248896 (n=4).

Cf. A382782.

cp's OEIS Frontend

A222068 Decimal expansion of (1/16)*Pi^2.

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A069075 a(n) = (4*n^2 - 1)^2.

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A248895 Decimal expansion of Sum_{i >= 1} 1/(4*i^2-1)^3.

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A248896 Decimal expansion of Sum_{i >= 1} 1/(4*i^2-1)^4.

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A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2*k) in the expansion of Sum_{k>=1} (1 / (4*k^2-1)^n).

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A382784 Irregular triangle T(n,k) read by rows of the coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n) with denominator 2^(2n)*(n-1)!.

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A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n).