A019669 -id:A019669 - cp's OEIS frontend

A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.

-1, 2, -16, 288, -9216, 460800, -33177600, 3251404800, -416179814400, 67421129932800, -13484225986560000, 3263182688747520000, -939796614359285760000, 317651255653438586880000, -124519292216147926056960000, 56033681497266566725632000000

Offset: 0

Eric W. Weisstein

sign

Coefficient of the Cayley-Menger determinant of order n.
A roller coaster has n rows of seats, each of which has room for two people.  |a(n)| is the number of ways n men and n women can be seated with a man and a woman in each row. - Geoffrey Critzer, Dec 17 2011
The o.g.f. of 1/a(n) is -BesselI(0,i*sqrt(2*x)), with i the imaginary unit. See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 10 2012
|a(n)|/2 is the number of integers k such that the digits of k and 2*k, written in base 2*n, are permutations of 0, 1, ..., 2*n-1. - Yifan Xie, Apr 12 2025

Andrew Howroyd, Table of n, a(n) for n = 0..100
Paul C. Kainen, Construction numbers: How to build a graph?, arXiv:2302.13186 [math.CO], 2023.
Usman A. Khan, Soummya Kar and Jose M. F. Moura, A novel geometric approach towards a linear theory for sensor localization, 2013.
Alan L. Mackay, On the regular heptagon, J. Math. Chemistry, vol. 21, 1997, 197-209.
Eric Weisstein's World of Mathematics, Cayley-Menger Determinant.

Cf. A000079, A001044, A019669, A334381, A334383.

Row of A340591 (in absolute values).

Mathematica
```
Table[(-1)^(n+1)2^n n!^2, {n, 0, 20}]
```

a(n)={(-1)^(n+1) * 2^n * n!^2} \\ Andrew Howroyd, Nov 07 2019

E.g.f.: -arcsinh(x/sqrt(2))^2. - Vladeta Jovovic, Aug 30 2004
Sum_{n>=0} |a(n)|/(2*n+1)! = Pi/2. - Daniel Suteu, Feb 06 2017
a(n) = (-1)^(n+1) * A000079(n) * A001044(n). - Terry D. Grant, May 21 2017
From Amiram Eldar, Nov 18 2020: (Start)
Sum_{n>=0} 1/a(n) = (-1) * A334383.
Sum_{n>=0} (-1)^(n+1)/a(n) = A334381. (End)

Terms a(14) and beyond from Andrew Howroyd, Nov 07 2019

A162330 Blocks of 4 numbers of the form 2k, 2k-1, 2k, 2k+1, k=1,2,3,4,...

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Jul 01 2009

This illustrates the infinite product Pi/2 = Product_{k>=1} ((2*k)/(2k-1))*((2k)/(2k+1)): read the 4 terms of numerator and denominator of the factor in the product in that order shown.

Number of roots of the polynomial 1+x+x^2+...+x^(n+1) = (x^(n+2)-1)/(x-1) in the left half plane. - Michel Lagneau, Oct 30 2012

Eric Weisstein's World of Mathematics, Wallis Formula.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000796, A019669, A134967, A183041.

Flatten[#+{0,-1,0,1}&/@Range[2,40,2]] (* Harvey P. Dale, Aug 12 2014 *)

A162330(n)=n+1-(n+2)\4*2  \\ M. F. Hasler, Nov 01 2012

a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x*(2-x+x^2+x^3-x^4)/((1+x)*(1+x^2)*(1-x)^2).
a(n) = n + 1 - 2*floor( (n+2)/4 ). - M. F. Hasler, Nov 01 2012
a(n) = (2*n + 3 - (-1)^n + 2*(-1)^((2*n - 1 + (-1)^n)/4))/4. - Luce ETIENNE, Mar 08 2016
Sum_{n>=1} (-1)^n/a(n) = 2*log(2) - 1. - Amiram Eldar, Sep 10 2023

Edited by R. J. Mathar, Sep 16 2009

A163746 Expansion of (theta_3(q)^2 + 3 * theta_3(q^3)^2) / 4 - 1 in powers of q.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 0, 1, 1, 2, 0, 3, 2, 0, 6, 1, 2, 1, 0, 2, 0, 0, 0, 3, 3, 2, 3, 0, 2, 6, 0, 1, 0, 2, 0, 1, 2, 0, 6, 2, 2, 0, 0, 0, 2, 0, 0, 3, 1, 3, 6, 2, 2, 3, 0, 0, 0, 2, 0, 6, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 9, 0, 0, 6, 0, 2, 1, 2, 0, 0, 4, 0, 6, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2, 1, 0, 3, 2, 6, 0, 2, 0

Offset: 1

Michael Somos, Aug 03 2009

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

			G.f. = q + q^2 + 3*q^3 + q^4 + 2*q^5 + 3*q^6 + q^8 + q^9 + 2*q^10 + 3*q^12 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A019669, A122856, A122865, A125061, A138741.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + 3 EllipticTheta[ 3, 0, q^3]^2) / 4 - 1, {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, e%2*2 + 1, p%4==1, e+1, 1-e%2)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^2) - 1, n))};

Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) - 1 in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) - 1 in powers of q.
Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1+(-1)^e)/2 if p == 3 (mod 4). [corrected by Amiram Eldar, Nov 14 2023]
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
a(n) = A125061(n) unless n=0. a(12*n + 7) = a(12*n + 11) = 0.
a(2*n) = a(n). a(2*n + 1) = A138741(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). - Michael Somos, Sep 02 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Nov 14 2023

A236367 Dihedral angle in a regular icosahedron (radians).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 23 2014

			2.41186499736282687500784672346618218880066348532739213...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.

Cf. A001622, Platonic solids dihedral angles: A137914 (tetrahedron), A156546 (octahedron), A019669 (cube), A137218 (dodecahedron).

RealDigits[2 * ArcTan[GoldenRatio^2], 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)

PARI
```
2*atan((3+sqrt(5))/2)
```

Equals 2*arctan(phi^2) = 2*arctan(A001622^2) = 2*arctan((3+sqrt(5))/2).

A236555 Decimal expansion of the steradian solid angle subtended by one triangular facet of the cuboctahedron.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Jan 28 2014

Also the vertex solid angle of a regular tetrahedron.

			0.55128559843253080794214415146445924293385062869017869619170040868416...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Cuboctahedron
Eric Weisstein's World of Mathematics, Spherical Polygon
Wikipedia, Platonic solid
Index entries for transcendental numbers.

Cf. A236556. Icosidodecahedron: A319881, A319883. Vertex Angles: A019669, A236557, A236558.

RealDigits[N[ArcCos[23/27], 107]][[1]] (* T. D. Noe, Jan 29 2014 *)

acos(23/27) \\ Charles R Greathouse IV, Feb 10 2025

arccos(23/27).
3*arcsin(2*sqrt(2)/3) - Pi. - Bradley Klee, Oct 04 2018
8*A236555 + 6*A236556 = 4*Pi. - Bradley Klee, Oct 04 2018
Equals 2*arctan(sqrt(2)/5). - R. J. Mathar, Mar 11 2025

Definition corrected by Bradley Klee, Oct 04 2018

A244978 Decimal expansion of Pi/32.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 09 2014

			0.0981747704246810387019576057274844651311615437304720569054670185096...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Beta Function
Eric W. Weisstein, Euler's Series Transformation.
Index entries for transcendental numbers

Cf. A019683, A244976, A244977, A019675, A000796, A003881, A019669, A019670, A019683.

Join[{0}, RealDigits[Pi/32, 10, 105] // First]

Pi/32 \\ Charles R Greathouse IV, Sep 28 2022

Equals Integral_{x = 0..1} x^2/(1 + x^2)^3 dx.
Also equals beta(3/2, 1/2)/16, where 'beta' is Euler's beta function.
From Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^4*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^5*sqrt(1 - x^4) dx = Integral_{x = 0..1} x^7*sqrt(1 - x^16) dx.
Equals Integral_{x >= 0} x^4/(1 + x^2)^4 dx. (End)
From Amiram Eldar, Jul 13 2020: (Start)
Equals Integral_{x=0..oo} dx/(x^2 + 4)^2.
Equals Sum_{k>=1} sin(k)^3*cos(k)^3/k. (End)
From  Peter Bala, Dec 08 2021: (Start)
Pi/32 = Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9)).
Applying Euler's series transformation to this alternating sum gives
Pi/32 = Sum_{n >= 1} 2^(n-3)*n*(n+1)/((2*n+3)*binomial(2*n+2, n+1)). (End)

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 0, 4, 1, 4, 0, 0, 2, 0, 0, 5, 2, 2, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 6, 0, 4, 0, 3, 2, 0, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 6, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 10, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 9, 2, 0, 0, 8, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002175, A008441, A019669, A053692, A113407, A121444.
Inverse Moebius transform of A034947.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[n_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)

{a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */

a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022

A036793 Decimal expansion of (2/Pi)*Integral_{x=0..Pi} sin(x)/x dx.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Integral(sin(x)/x dx) = x - x^3/(3*3!) + x^5/(5*5!) - x^7/(7*7!) + ... . - Harry J. Smith, Apr 28 2009

			1.17897974447216727..., the constant in Gibbs phenomenon.

E. J. Borowski and J. M. Borwein, Dictionary of Mathematics, 3rd printing, Harper Collins, 1991, Gibbs phenomenon.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 249.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Wilbraham-Gibbs Constant

Cf. A036791 (continued fraction), A061079 for Si( x ).

RealDigits[ N[ (2/Pi)*SinIntegral[Pi], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

{ default(realprecision, 20080); y=0; x=Pi; m=x; x2=x*x; n=1; nf=1; s=1; while (x!=y, y=x; n++; nf*=n; n++; nf*=n; m*=x2; s=-s; x+=s*m/(n*nf)); x*=2/Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b036793.txt", n, " ", d)); } \\ Harry J. Smith, Apr 28 2009

A036792 divided by A019669. - R. J. Mathar, Mar 22 2011

A236556 Decimal expansion of the steradian solid angle subtended by one square facet of the cuboctahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 28 2014

Also the vertex solid angle of a regular octahedron.

			1.35934763781648774838557005356705626555297876132983228572769584995966...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Cuboctahedron.
Eric Weisstein's World of Mathematics, Spherical Polygon.
Wikipedia, Platonic solid.
Index entries for transcendental numbers.

Cf. A188615, A236555. Icosidodecahedron: A319881, A319883. Vertex Angles: A019669, A236557, A236558.

evalf(4*arcsin(1/3),100) ; # R. J. Mathar, Apr 26 2021

RealDigits[4*ArcSin[1/3], 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

PARI
```
4*asin(1/3)
```

Equals 4*arcsin(1/3) = 4*A188615.
Equals 2*Pi - 4*arcsin(2*sqrt(2)/3). - Bradley Klee, Oct 04 2018
8*A236555 + 6*this = 4*Pi. - Bradley Klee, Oct 04 2018

Definition corrected by Bradley Klee, Oct 04 2018

A094888 Decimal expansion of 2Piphi, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Jun 15 2004

			10.16640738463051963161901802648439768366367858644230824...

Ivan Panchenko, Table of n, a(n) for n = 2..1000
John Baez, Pi and the Golden Ratio, Azimuth, Mar 07 2017.
Index entries for transcendental numbers.

Cf. A000796, A001622, A090771, A094886, A135155.

Integral_{x>=0} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), A019670 (m=6), A352125 (m=8), this sequence (m=10).

evalf(Pi*(1+sqrt(5)), 121);  # Alois P. Heinz, May 16 2022

RealDigits[2 * Pi * GoldenRatio, 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

From Peter Bala, Nov 03 2019: (Start)
Equals 10*Integral_{x >= 0} cosh(4*x)/cosh(5*x) dx = Integral_{x = 0..1} (1 + x^8)/(1 + x^10) dx .
Equals 100*Sum_{n >= 0} (-1)^n*(2*n + 1)/( (10*n + 1)*(10*n + 9) ). (End)
Equals 10 * Product_{k>=2} 2/sqrt(2 + sqrt(2 + ... sqrt(2 + phi)...)), with k nested radicals (Baez, 2017). - Amiram Eldar, May 18 2021
Equals Integral_{x>=0} 1/(1 + x^10) dx = (Pi/10) * csc(Pi/10). - Bernard Schott, May 15 2022
Equals Gamma(1/10)*Gamma(9/10). - Andrea Pinos, Jul 03 2023
Equals 10 * Product_{k >= 1} (10*k)^2/((10*k)^2 - 1). - Antonio Graciá Llorente, Mar 15 2024
Equals 10 * Product_{k>=2} (1 + (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A094886 = 10*A135155/e. - Hugo Pfoertner, Nov 23 2024

cp's OEIS Frontend

A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.

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A162330 Blocks of 4 numbers of the form 2k, 2k-1, 2k, 2k+1, k=1,2,3,4,...

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A163746 Expansion of (theta_3(q)^2 + 3 * theta_3(q^3)^2) / 4 - 1 in powers of q.

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A236367 Dihedral angle in a regular icosahedron (radians).

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A236555 Decimal expansion of the steradian solid angle subtended by one triangular facet of the cuboctahedron.

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PARI

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A244978 Decimal expansion of Pi/32.

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Mathematica

PARI

Formula

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

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A094888 Decimal expansion of 2Piphi, where phi = (1+sqrt(5))/2.