A003881 -id:A003881 - cp's OEIS frontend

A004711 Positions of 1's in binary expansion of Pi/4.

1, 2, 5, 8, 13, 14, 15, 16, 17, 18, 20, 21, 23, 25, 27, 31, 35, 40, 42, 43, 45, 49, 50, 55, 59, 60, 62, 65, 66, 70, 73, 74, 78, 79, 82, 83, 87, 89, 93, 95, 96, 97, 105, 106, 108, 109, 110, 116, 117, 118, 121, 122, 124

Offset: 1

Simon Plouffe

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003881, A004601, A256108.

Flatten[Position[RealDigits[Pi/4,2,150][[1]],1]] (* Harvey P. Dale, Jul 13 2015 *)

default(realprecision, 100); select(x->(x==1), binary(Pi/4)[2], 1) \\ Michel Marcus, Oct 11 2018

a(n) = A256108(n) + 2. - Lorenzo Sauras Altuzarra, Jan 21 2020

A134013 Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 4, 0, 0

Offset: 1

Michael Somos, Oct 02 2007

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + 2*q^2 + 2*q^5 + q^9 + 4*q^10 + 2*q^13 + 2*q^17 + 2*q^18 + 3*q^25 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A003881, A008441, A053692, A112301, A113407, A134014.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^4], {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n>0 && (n+1)%4\2, (n%4) * sumdiv( n/gcd(n,2), d, (-1)^(d\2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^8 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^16)^2 / ( eta(q)^2 * eta(q^4)^2 * eta(q^8) ) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, ...].
Moebius transform is period 16 sequence [ 1, 1, -1, -2, 1, -1, -1, 0, 1, 1, -1, 2, 1, -1, -1, 0, ...].
a(n) is multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134014.
a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = 2 * a(4*n + 1).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A112301(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n = 5) = 2 * A053692(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 (A003881). - Amiram Eldar, Nov 24 2023

A141904 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of some polynomials P(n,x).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 23, -1, 1, 1, -44, 14, -4, 1, -1, 563, -818, 22, -5, 1, 1, -3254, 141, -1436, 19, -2, 1, -1, 88069, -13063, 21757, -457, 43, -7, 1, 1, -11384, 16774564, -11368, 7474, -680, 56, -8, 1, -1, 1593269, -1057052, 35874836, -261502, 3982, -688, 212, -3, 1, 1, -15518938, 4651811

Offset: 0

Paul Curtz, Sep 14 2008

Let the polynomials P be defined by P(0,x)=u(0), P(n,x)= u(n) + x*sum_{i=0..n-1} u(i)*P(n-i-1,x) and coefficients u(i)=(-1)^i/(2i+1). These u are reminiscent of the Leibniz' Taylor expansion to calculate arctan(1) =pi/4 = A003881. Then P(n,x) = sum_{k=0..n} c(n,k)*x^k.

			The polynomials P(n,x) are for n=0 to 5:
1 = P(0,x).
-1/3+x = P(1,x).
1/5-2/3*x+x^2 = P(2,x).
-1/7+23/45*x-x^2+x^3 = P(3,x).
1/9-44/105*x+14/15*x^2-4/3*x^3+x^4 = P(4,x).
-1/11+563/1575*x-818/945*x^2+22/15*x^3-5/3*x^4+x^5 = P(5,x).

P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p.44.
P. Flajolet, X. Gourdon, B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp.67-78.

Jean-François Alcover, Roots of P(120,x) in the shape of a cardioid.

Cf. A142048 (denominators), A140749, A141412 (where u=(-1)^i/(i+1)).

u := proc(i) (-1)^i/(2*i+1) ; end:
P := proc(n,x) option remember ; if n =0 then u(0); else u(n)+x*add( u(i)*procname(n-1-i,x),i=0..n-1) ; expand(%) ; fi; end:
A141904 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end: seq(seq(A141904(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Aug 24 2009

ClearAll[u, p]; u[n_] := (-1)^n/(2*n + 1); p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n - i - 1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[ p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Numerator (* Jean-François Alcover, Oct 02 2012 *)

Edited and extended by R. J. Mathar, Aug 24 2009

A151842 a(3n) = n, a(3n+1) = 2n+1, a(3n+2) = n+1.

Original entry on oeis.org

0, 1, 1, 1, 3, 2, 2, 5, 3, 3, 7, 4, 4, 9, 5, 5, 11, 6, 6, 13, 7, 7, 15, 8, 8, 17, 9, 9, 19, 10, 10, 21, 11, 11, 23, 12, 12, 25, 13, 13, 27, 14, 14, 29, 15, 15, 31, 16, 16, 33, 17, 17, 35, 18, 18, 37, 19, 19, 39, 20, 20, 41, 21, 21, 43, 22, 22, 45, 23, 23, 47

Offset: 0

Shane Geiger (shane.geiger(AT)gmail.com), Jul 14 2009

Take a list of numbers (like 0,1,2,3,4,5,...) and then pair them up like this: (0,1)(1,2),(2,3),(3,4)... Then sum each pair, and insert the sum between the numbers, like this: (0,1,1), (1,3,2), (2,5,3), ... Finally, remove the parentheses: 0,1,1,1,3,2,2,5,3,...
This mirrors the pattern used to make a dragon curve fractal. You take two points, then find one to insert between them. In the next iteration, you take those three points and find two numbers to insert between them. (Rather than summing the two numbers, a different function is used to find a point relative to two other points.)
a(n) is the number of rises in all compositions of n + 2 with parts in {1,2} and adjacent differences in {-1,1}. - John Tyler Rascoe, Apr 29 2025

			G.f. = x + x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 5*x^7 + 3*x^8 + 3*x^9 + ... - _Michael Somos_, Aug 12 2009

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

See A076118 for a version with signs.

Cf. A002264, A003881, A008133.

I:=[0,1,1,1,3,2]; [n le 6 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..80]]; // Vincenzo Librandi, Feb 14 2015

CoefficientList[Series[x (1 + x) (1 + x^2) / ((x - 1)^2 (1 + x + x^2)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Feb 14 2015 *)

{a(n) = kronecker(9, n) + (n\3) * [1, 2, 1][n%3 + 1]} /* Michael Somos, Aug 12 2009 */

def pairup(x): return [x[i:i+2] for i in range(len(x)-1)]
def combine(vals): return sum(vals)
def expand(L,fn): return [(x[0],fn(x),x[1]) for x in pairup(L)]
L = list(range(20))
print(expand(L,combine))

From R. J. Mathar, Jul 14 2009: (Start)
G.f.: x*(1+x)*(1+x^2)/((x-1)^2*(1+x+x^2)^2).
a(n) = 2*a(n-3) - a(n-6). (End)
From Michael Somos, Aug 12 2009: (Start)
G.f.: x * (1 - x^4) / ((1 - x) * (1 - x^3)^2).
Euler transform of length 4 sequence [ 1, 0, 2, -1]. (End)
-a(n) = a(-1-n). - Michael Somos, Nov 11 2013
From Ridouane Oudra, Nov 23 2024: (Start)
a(n) = 5*n/6 + n^2/2 - n^3/3 + (2*n^2 - n - 3/2)*floor(n/3) - (3*n + 3/2)*floor(n/3)^2.
a(n) = t(n+2)*t(n+3) - t(n)*t(n+1), where t(n) = floor(n/3) = A002264(n).
a(n) = A008133(n+2) - A008133(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - Amiram Eldar, May 10 2025

More terms from Vincenzo Librandi, Feb 14 2015

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

A243445 Decimal expansion of the polar angle of the cone circumscribed to a regular dodecahedron from one of its vertices.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jun 06 2014

The angle is in radians.

			1.20593249868141343750392336173309109440033174266369606513299755...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Dodecahedron (use the point coordinates to derive the formula).

Cf. A001622 (phi), A003881 (octahedron), A195695 (tetrahedron), A195696 (cube), A195723 (isosahedron).

RealDigits[ArcCos[1/(GoldenRatio Sqrt[3])],10,120][[1]] (* Harvey P. Dale, May 17 2016 *)

PARI
```
acos(2/(1+sqrt(5))/sqrt(3))
```

arccos(1/(phi*sqrt(3))), where phi = A001622.

arctan(phi^2), where phi = A001622. - Jon Maiga, Nov 11 2018

A261624 Decimal expansion of the Dirichlet beta function at 1/5.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 27 2015

			0.57371084718594664935726652783200417043624693824269093761895362825...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Dirichlet Beta Function
Wikipedia, Dirichlet beta function

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A261622 (beta(1/3)), A261623 (beta(1/4)).

evalf(Sum((-1)^n/(2*n+1)^(1/5), n=0..infinity), 120); # Vaclav Kotesovec, Aug 27 2015

RealDigits[DirichletBeta[1/5],10,102]//First

beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
beta(.2) \\ Charles R Greathouse IV, Oct 18 2024

beta(1/5) = (zeta(1/5, 1/4) - zeta(1/5, 3/4))/2^(2/5).

A387322 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and a square face at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.4712905456469785754732547961552537994857493330886...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square bicupola.
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square bicupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A003881, A002193, A010487.

First[RealDigits[Pi/4 + ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J23", "DihedralAngles"]], 4], 10, 100]]

Equals Pi/4 + arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = A003881 + arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A003881 + A387323.

A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and the octagonal face.

			1.6858923822495302658575939503353780784364569832448...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387322.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A002193, A003881, A010487, A195696.

First[RealDigits[ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A387321 - A195696 = A387322 - A003881.

A105532 Decimal expansion of arctan(1/5).

Original entry on oeis.org

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

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

			0.197395559849880758370049765194790293447585103787852101517688940241033969...

Eric Weisstein's World of Mathematics, Machin-Like Formulas
Index entries for transcendental numbers

Cf. A003881 (Pi/4), A072172, A105534 (arctan 1/239).

RealDigits[ArcTan[1/5], 10, 100][[1]] (* Amiram Eldar, Aug 04 2020 *)

PARI
```
atan(1/5) \\ Michel Marcus, Sep 24 2014
```

4*arctan(1/5) - A105534 = Pi/4 (Machin's formula).
From Amiram Eldar, Aug 04 2020: (Start)
Equals Sum_{k>=0} (-1)^k/((2*k+1) * 5^(2*k+1)) = Sum_{k>=0} (-1)^k/A072172(k).
Equals Sum_{k>=1} arctan(1/(2*(k+2)^2)). (End)

Corrected position of decimal point in example. - R. J. Mathar, Feb 05 2009

cp's OEIS Frontend

A004711 Positions of 1's in binary expansion of Pi/4.

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A134013 Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

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A141904 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of some polynomials P(n,x).

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A151842 a(3n) = n, a(3n+1) = 2n+1, a(3n+2) = n+1.

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

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A243445 Decimal expansion of the polar angle of the cone circumscribed to a regular dodecahedron from one of its vertices.

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A261624 Decimal expansion of the Dirichlet beta function at 1/5.

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