A086089 -id:A086089 - cp's OEIS frontend

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

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

Offset: 1

Bruno Berselli, Mar 06 2015

Value of the Borwein-Borwein function I_3(a,b) for a = b = 1. - Stanislav Sykora, Apr 16 2015

The area of a circle circumscribing a unit-area regular hexagon. - Amiram Eldar, Nov 05 2020

			1.2091995761561452337293855050947704881893774987284937170465899569254...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), pp. 120-121.
L. B. W. Jolley, Summation of Series, Dover (1961), No. 261, pp. 48, 49, (and No. 275).

Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).
Su Hu and Min-Soo Kim, A generalization of Wallis' formula, arXiv:2201.09674 [math.NT], 2022.
Richard Kershner, The Number of Circles Covering a Set, American Journal of Mathematics, 61(3), 665-671.
MIT Integration Bee, 2023 MIT Integration Bee - Finals, Problem 1.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), C6.2.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See (9) p. 146.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean, equations 26-32.
Index entries for transcendental numbers

Cf. A000796, A001651, A002194, A093602, A248181, A257096, A257097, A186706.
Cf. A010815, A073010, A086089, A214549, A256923, A275486.
Cf. A091682 (Sum_{i >= 0} (i!)^2/(2*i)!).

RealDigits[2 Sqrt[3] Pi/9, 10, 100][[1]]

a = 2*Pi/(3*sqrt(3)) \\ Stanislav Sykora, Apr 16 2015

Equals 2*sqrt(3)*Pi/9 = 1 + 1/6 + 1/30 + 1/140 + 1/630 + 1/2772 + 1/12012 + ...
Equals m*I_3(m,m) = m*Integral_{x>=0} (x/(m^3+x^3)), for any m>0. - Stanislav Sykora, Apr 16 2015
Equals Integral_{x>=0} (1/(1+x^3)) dx. - Robert FERREOL, Dec 23 2016
From Peter Bala, Oct 27 2019: (Start)
Equals 3/4*Sum_{n >= 0} (n+1)!*(n+2)!/(2*n+3)!.
Equals Sum_{n >= 1} 3^(n-1)/(n*binomial(2*n,n)).
Equals 2*Sum_{n >= 1} 1/(n*binomial(2*n,n)). See Boros and Moll, pp. 120-121.
Equals Integral_{x = 0..1} 1/(1 - x^3)^(1/3) dx = Sum_{n >= 0} (-1)^n*binomial(-1/3,n) /(3*n + 1).
Equals 2*Sum_{n >= 1} 1/((3*n-1)*(3*n-2)) = 2*(1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...) (added Oct 30 2019). (End)
Equals Product_{k>=1} 9*k^2/(9*k^2 - 1). - Amiram Eldar, Aug 04 2020
From Peter Bala, Dec 13 2021: (Start)
Equals (2/3)*A093602.
Conjecture: for k >= 0, 2*sqrt(3)*Pi/9 = (3/2)^k * k!*Sum_{n = -oo..oo} (-1)^n/ Product_{j = 0..k} (3*n + 3*j + 1). (End)
Equals (3/4)*S - 1, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..Pi/2} tan(x)^(1/3)/(sin(2*x) + 1) dx. See MIT Link. - Joost de Winter, Aug 26 2023
Continued fraction: 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). See A000407. - Peter Bala, Feb 20 2024
Equals Sum_{n>=2} 1/binomial(n, floor(n/2)); and trivially if "floor" is replaced by "ceiling". - Richard R. Forberg, Aug 30 2024
Equals Product_{k>=2} (1 + (-1)^k/A001651(k)). - Amiram Eldar, Nov 22 2024
Equals 2*A073010 = 1/A086089 = sqrt(A214549) = exp(A256923) = A275486/2. - Hugo Pfoertner, Nov 22 2024
Equals 1 - (1/6) * Sum_{n>=1} A010815(n)/n. - Friedjof Tellkamp, Apr 05 2025
Equals A248181 - 2. - Pontus von Brömssen, Apr 05 2025

A016766 a(n) = (3*n)^2.

Original entry on oeis.org

0, 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296, 1521, 1764, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 4356, 4761, 5184, 5625, 6084, 6561, 7056, 7569, 8100, 8649, 9216, 9801, 10404, 11025, 11664, 12321, 12996, 13689, 14400, 15129, 15876, 16641, 17424

Offset: 0

N. J. A. Sloane

Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - Roberto E. Martinez II, Jan 07 2002
Area of a square with side 3n. - Wesley Ivan Hurt, Sep 24 2014
Right-hand side of the binomial coefficient identity Sum_{k = 0..3*n} (-1)^(n+k+1)* binomial(3*n,k)*binomial(3*n + k,k)*(3*n - k) = a(n). - Peter Bala, Jan 12 2022

Ivan Panchenko, Table of n, a(n) for n = 0..200
John Elias, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Numbers of the form 9*n^2 + k*n, for integer n: this sequence (k = 0), A132355 (k = 2), A185039 (k = 4), A057780 (k = 6), A218864 (k = 8). - Jason Kimberley, Nov 09 2012

Cf. A000217, A000290, A008585, A033428, A051624, A086089, A195042.

[(3*n)^2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2014

A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # Wesley Ivan Hurt, Sep 24 2014

(3Range[0, 49])^2 (* Alonso del Arte, Sep 24 2014 *)

A016766(n):=(3*n)^2$
makelist(A016766(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=9*n^2 \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 9*n^2 = 9*A000290(n). - Omar E. Pol, Dec 11 2008
a(n) = 3*A033428(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 9*(2*n-1) for n > 0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
From Wesley Ivan Hurt, Sep 24 2014: (Start)
G.f.: 9*x*(1 + x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = A000290(A008585(n)). (End)
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/54.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).
Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)
a(n) = A051624(n) + 8*A000217(n).  In general, if P(k,n) = the k-th n-gonal number, then (k*n)^2 = P(k^2 + 3,n) + (k^2 - 1)*A000217(n). - Charlie Marion, Mar 09 2022
From Elmo R. Oliveira, Nov 30 2024: (Start)
E.g.f.: 9*x*(1 + x)*exp(x).
a(n) = n*A008591(n) = A195042(2*n). (End)

More terms from Zerinvary Lajos, May 30 2006

A240935 Decimal expansion of 3sqrt(3)/(4Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 03 2014

A triangle of maximal area inside a circle is necessarily an inscribed equilateral triangle. This constant is the ratio of the triangle's area to the circle's area. In general, the ratio of an arbitrary triangle's area to the area of its unique Steiner ellipse, which has the least area of any circumscribed ellipse (an equilateral triangle's Steiner ellipse is a circle).

Also the probability that the distance between 2 randomly selected points within a circle will be larger than the radius. - Amiram Eldar, Mar 03 2019

			0.4134966715663440371334948737347270810480...

B. F. Finkel, Problem 38, solved by O. W. Anthony, Henry Heaton, and G. B. M. Zerr, The American Mathematical Monthly, Vol. 3, No. 12 (1896), pp. 324-326.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), p.52
I. Todhunter, A treatise on the Integral Calculus, London and Cambridge: MacMillan and Co., 1868, page 320, Example 7.
Wikipedia, Steiner ellipse
Index entries for transcendental numbers

Cf. A073010, A060294, A003881, A002194, A000796, A102519, A086089.

Digits:=100: evalf(3*sqrt(3)/(4*Pi)); # Wesley Ivan Hurt, Aug 03 2014

Flatten[RealDigits[3 Sqrt[3]/(4 Pi), 10, 100, -1]] (* Wesley Ivan Hurt, Aug 03 2014 *)

default(realprecision, 120);
3*sqrt(3)/(4*Pi)

3*sqrt(3)/(4*Pi) = 3*A002194/(4*A000796).

Equals A093604^2. - Hugo Pfoertner, May 18 2024

A185652 Number of permutations of [n] having a shortest ascending run of length 2.

Original entry on oeis.org

0, 0, 1, 0, 5, 18, 89, 519, 3853, 27555, 233431, 2167152, 21596120, 232817282, 2718706924, 33814848445, 448311181346, 6319365554730, 94225534689624, 1481940898130323, 24536143182460549, 426432943716156580, 7762187693343502658, 147704506384475066381

Offset: 0

Alois P. Heinz, Aug 29 2013

nonn

			a(2) = 1: 12.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 18: 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 23145, 23415, 23514, 24135, 24513, 25134, 34125, 34512, 35124, 45123.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=2 of A064315.

Cf. A086089 (3*sqrt(3)/(2*Pi)).

A[n_, k_] := A[n, k] = Module[{b}, b[u_, o_, t_] := b[u, o, t] = If[t + o <= k, (u + o)!, Sum[b[u + i - 1, o - i, Min[k, t] + 1], {i, 1, o}] + If[t <= k, u (u + o - 1)!, Sum[b[u - i, o + i - 1, 1], {i, 1, u}]]]; Sum[b[j - 1, n - j, 1], {j, 1, n}]];
a[n_] := A[n, 2] - A[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 26 2021, after Alois P. Heinz in A064315 *)

a(n) ~ c * (3*sqrt(3)/(2*Pi))^n * n!, where c = 0.45178068752734823... . - Vaclav Kotesovec, Sep 06 2014

A306712 Decimal expansion of 3*sqrt(3)/Pi.

Original entry on oeis.org

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

Offset: 1

Scott R. Shannon, Mar 05 2019

This is the mean end-to-end distance of the 2-step self-avoiding walk with full excluded volume in the 2-dimensional continuum.
Take 3 touching circles of diameter 1 which are joined as a chain and each is free to move around its neighbors' perimeters, but no circle can overlap another. This value is the average of the distance from the middle of the first circle to the middle of the third circle, averaged over all possible configurations the chain of 3 non-overlapping circles can take.
Using the law of cosines one can show the distance between the middle of the first and third circles, r_3, in the 3-circle chain is r_3 = sqrt(2-2*cos(t)), where t is the angle between these circles centered on the second circle. The mean end-to-end distance is thus given by the integral  = Integrate(r_3,{t,Pi/3,5*Pi/3})/(4*Pi/3), which includes division by the required normalization constant. Solving this definite integral gives the exact value for  as 3*sqrt(3)/Pi. This is A289504 minus 2.
Removing the square root from r_3 in the above integral gives the mean square end-to-end distance for the 2-step walk. Evaluating this integral gives the exact value for  as 2+3*sqrt(3)/(2*Pi), with a value of approximately 2.826993343... . This is A086089 plus 2, or equivalently this sequence divided by 2, plus 2.

			1.653986686265376148533979494938908324192159441099921958398...

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Java code for a 2-step self-avoiding walk in the continuum. This produces a numerical approximation to the exact values.
Index entries for transcendental numbers.

Cf. A306648, A086089, A289504, A073010 (reciprocal).

RealDigits[3*Sqrt[3]/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

PARI
```
3*sqrt(3)/Pi
```

Terms a(59) and beyond from Andrew Howroyd, Apr 27 2020

A289504 Decimal expansion of 2(1+3^(3/2)/(2Pi)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jul 07 2017

			3.6539866862653761485339794949389083241921...

Steven R. Finch, Mean width of a regular simplex, arxiv:1111.4976 [math.MG], (2016), nu_2.
Index entries for transcendental numbers

Maple
```
2*(1+3*sqrt(3)/2/Pi); evalf(%) ;
```

RealDigits[2(1+3^(3/2)/(2Pi)),10,120][[1]] (* Harvey P. Dale, Nov 05 2019 *)

2+3*sqrt(3)/Pi \\ Charles R Greathouse IV, Oct 01 2022

from mpmath import *
mp.dps=86
C=2*(1 + 3*sqrt(3)/(2*pi))
print([int(n) for n in list(str(C).replace('.', ''))]) # Indranil Ghosh, Jul 08 2017

Equals 2*(1+A086089).

A371466 Decimal expansion of Product_{k>=1} (1 - 1/(3*k+1)^2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 31 2024

			0.8833193751427249786568447498242193512859342691...

Cf. A003881, A010503, A086089, A096427, A224268, A290570, A371467.

RealDigits[Gamma[1/3]^3/(4 Sqrt[3] Pi), 10, 100][[1]]

Equals Gamma(1/3)^3 / (4 * sqrt(3) * Pi).
Equals A290570/2. - Hugo Pfoertner, Mar 31 2024
Equals Integral_{x=0..1} (1-x^3)^(1/3) dx. - Mikhail Kurkov, Jun 29 2025

A371467 Decimal expansion of Product_{k>=0} (1 - 1/(3*k+2)^2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 31 2024

			0.6844634059797257270110769788663463289556838...

Cf. A003881, A010503, A086089, A096427, A224268, A224273, A371466.

RealDigits[(4/3) Pi^2/Gamma[1/3]^3, 10, 100][[1]]

Equals (4/3) * Pi^2 / Gamma(1/3)^3.

Equals 1/A224273. - Hugo Pfoertner, Mar 31 2024

A371604 Decimal expansion of 5 * sqrt(3 - phi) / (2 * Pi).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 01 2024

			0.93548928378863903321291906615298281...

Cf. A019845, A060294, A086089, A089491, A094874, A112628, A182007, A258403.

RealDigits[5 Sqrt[3 - GoldenRatio]/(2 Pi), 10, 99][[1]]

Equals Product_{k>=1} (1 - 1/(5*k)^2).

Equals A258403/Pi. - Hugo Pfoertner, Apr 01 2024

A241624 Decimal expansion of 'c', one of Polya's Random Walk constants, related to the asymptotics of the number of 3-D random walks starting from and returning to the origin.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 19 2014

			0.53923817508158141965602919129297773...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.9 p. 325.

Cf. A086089, A086231.

m3 = A086231 = Sqrt[6]/32/Pi^3*Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]; c = 9/32*(m3 + 6/(Pi^2*m3)); RealDigits[c, 10, 100] // First

c = 9/32*(m3 + 6/(Pi^2*m3)), where m3 = A086231.

cp's OEIS Frontend

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

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A016766 a(n) = (3*n)^2.

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A240935 Decimal expansion of 3*sqrt(3)/(4*Pi).

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A185652 Number of permutations of [n] having a shortest ascending run of length 2.

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A306712 Decimal expansion of 3*sqrt(3)/Pi.

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A289504 Decimal expansion of 2*(1+3^(3/2)/(2*Pi)).

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A371466 Decimal expansion of Product_{k>=1} (1 - 1/(3*k+1)^2).

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A240935 Decimal expansion of 3sqrt(3)/(4Pi).

A289504 Decimal expansion of 2(1+3^(3/2)/(2Pi)).