A019670 -id:A019670 - cp's OEIS frontend

A244978 Decimal expansion of Pi/32.

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)

A258256 Expansion of f(q^3) * psi(-q^3)^3 / (psi(-q) * psi(-q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 1, 2, 4, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 4, 2, 0, 0, 2, 2, 0, 0, 0, 8, 0, 0, 0, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 4, 0

Offset: 0

Michael Somos, May 24 2015

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

			G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 + 4*q^9 + 2*q^10 + 2*q^13 + q^16 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A019670, A089801, A121444, A122856, A122856, A256282, A258260.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + A[2] + A[3] + A[5] + 2*A[6] + A[9] + 4*A[10] + 2*A[11] + 2*A[14] + A[17] + 2*A[18] + 4*A[19];

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[ #, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + n + #) &]];
a[ n_] := If[ n < 2, Boole[n >= 0], Times @@ (Which[ # == 2, 1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)]), {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

Expansion of eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18) / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, ...].
Moebius transform is period 36 sequence [1, 0, -1, 0, 1, 0, -1, 0, 4, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -4, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
a(2*n) = a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 3) = 0. a(12*n + 1) = A002175. a(12*n + 5) = 2 * A121444(n).
a(n) = Sum_{d|n} A258260(d) * (-1)^(n+d) if n>0.
a(n) = (-1)^n * A256282(n). - Michael Somos, Jun 06 2015
a(n) is multiplicative with a(0) = 1, a(2^e) = 1, a(3^e) = 2*(1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015
Expansion of A0(x)^2 + A0(x)*A1(x) + A1(x)^2 in powers of x where A0(x) = phi(x^9), A1(x) = x * f(x^3, x^15) = x * A089801(x^3). - Michael Somos, Jun 23 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Nov 24 2023

A122864 Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 15 2006

			q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + q^8 - 2*q^9 + 2*q^10 + 2*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002564, A019670, A035154, A113446, A122856, A122865.

eta[x_] := x^(1/24)*QPochhammer[x]; A122864[n_] := SeriesCoefficient[ (eta[q^3]^2*eta[q^4]*eta[q^6]^2*eta[q^36])/(eta[q]*eta[q^9]*eta[q^12]^2), {q, 0, n}]; Table[A122864[n], {n, 50}] (* G. C. Greubel, Sep 16 2017 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^36 + A) /(eta(x + A) * eta(x^9 + A) * eta(x^12 + A)^2), n))}

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

Euler transform of period 36 sequence [ 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, 1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, 1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2*(-1)^(e+1) if e>0, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
a(3*n) = 2 * A113446(n). a(3*n + 1) = A002564(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Oct 15 2022

A281451 Expansion of x * f(x, x) * f(x, x^17) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 23 2017

nonn

			G.f. = x + 3*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + 2*x^10 + 2*x^11 + 2*x^17 + ...
G.f. = q^16 + 3*q^25 + 2*q^34 + 2*q^52 + 2*q^61 + 2*q^97 + 2*q^106 + ...

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. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
Cf. A258228, A258256, A258278, A258292, A259761.
Cf. A281452, A281453, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 7, KroneckerSymbol[ -4, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 7])];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^18] QPochhammer[ -x^17, x^18] QPochhammer[ x^18], {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv(9*n + 7, d, (d%4==1) - (d%4==3)))};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 7); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};

{a(n) = if( n<0, 0, my(m = 9*n + 7, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 4 || k%9 == 5), s+=(j>0)+1)); s)};

f(x,x^m) = 1 + Sum_{k>=1} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 8*k)).
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-17)) * (1 + x^(18*k-1)) * (1 - x^(18*k)).
a(4*n) = a(8*n + 7) = a(16*n + 13) = a(32*n + 9) = a(49*n + 7) = a(98*n + 14) = 0.
a(4*n + 1) = A281452(n).  a(8*n + 3) = 2 * A281491(n).  A(16*n + 1) = A281453(n).
a(32*n + 25) = 2 * A281490(n). a(64*n + 49) = a(n). a(128*n + 17) = 2 * A281492(n).
a(n) = A122865(3*n + 2). a(n) = A122856(6*n + 4) = A258278(6*n + 4).
a(n) = b(9*n + 7) where b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 7) where b = A105673, A122857, A258034, A259761.  -2 * a(n) = b(9*n + 7) where b = A138949, A256280, A258292.
a(n) = - A256269(9*n + 7). 4 * a(n) = A004018(9*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

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

A164105 Decimal expansion of Pi^3/6.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 6-dimensional unit sphere.

			Equals 5.1677127800499700292460525111835658670375480943...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164106, A164108.

RealDigits[Pi^3/6, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^3/6 \\ G. C. Greubel, Apr 09 2017
```

Equals A091925/6 = A019670*A102753.

A258034 Expansion of phi(q) * phi(q^9) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 4, 4, 0, 0, 4, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 8, 0

Offset: 0

Michael Somos, Jun 03 2015

nonn

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

			G.f. = 1 + 2*q + 2*q^4 + 4*q^9 + 4*q^10 + 4*q^13 + 2*q^16 + 4*q^18 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A004018, A019670, A028601, A122856, A122865, A258322.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[19];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9], {q, 0, n}];
a[ n_] := Which[ n < 1, Boole[n == 0], Mod[n, 3] == 2, 0, True, 2 DivisorSum[ n, If[ Mod[n/#, 9] > 0, 1, 2] KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jul 04 2015 *)

{a(n) = if( n<1, n==0, (n+1)%3 * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A))^2, n))};

{a(n) = if( n<1, n==0, n%3==2, 0, 2 * sumdiv(n, d, if(n\d%9, 1, 2) * kronecker( -4, d)))}; /* Michael Somos, Jul 04 2015 */

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); (n%3 < 2) * 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Jul 04 2015 */

Expansion of eta(q^2)^5 * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36))^2 in powers of q.
Euler transform of period 36 sequence [2, -3, 2, -1, 2, -3, 2, -1, 4, -3, 2, -1, 2, -3, 2, -1, 2, -6, 2, -1, 2, -3, 2, -1, 2, -3, 4, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A258322(n). a(4*n) = a(n).
a(3*n + 2) = a(4*n + 3) = a(8*n + 6) =  a(9*n + 3) = a(9*n + 6) = 0.
a(3*n + 1) = 2 * A122865(n).  a(6*n + 4) = 2 * A122856(n). a(9*n) = A004018(n). a(12*n + 1) = 2 * A002175(n).
a(2*n) = A028601(n). - Michael Somos, Jul 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Jan 29 2024

A352125 Decimal expansion of Pisqrt(2)sqrt(2 + sqrt(2))/8.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Mar 05 2022

			1.02617215297703088887146778087283197497962...

Jean-François Pabion, Éléments d'Analyse Complexe, licence de Mathématiques, page 111, Ellipses, 1995.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, pp. 235-236.
Index entries for transcendental numbers.

Cf. A000796, A047522, A121601.

Integral_{x=0..oo} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), A019670 (m=6), this sequence (m=8), A094888 (m=10).

First[RealDigits[N[Pi*Sqrt[2]Sqrt[2+Sqrt[2]]/8,95]]]

Pi*sqrt(4 + 2*sqrt(2))/8 \\ Michel Marcus, Mar 07 2022

Equals Integral_{x=0..oo} 1/(1 + x^8) dx.
Equals Pi*csc(Pi/8)/8.
Equals 1/Product_{k>=1} (1 - 1/(8*k)^2). - Amiram Eldar, Mar 12 2022
Equals Product_{k>=2} (1 + (-1)^k/A047522(k)). - Amiram Eldar, Nov 22 2024

A019691 Decimal expansion of Pi/24.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With a different offset, also decimal expansion of 5*Pi/12, 25*Pi/6 or 125*Pi/3. - Michel Marcus, Sep 09 2013

Volume of a quarter sphere of diameter 1. - Omar E. Pol, Aug 19 2019

			0.13089969389957471826927680763664595350821539164062940920728935801282...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. A000796, A019670, A019673, A019675, A019679.

RealDigits[N[Pi/24,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

Pi/24 \\ Charles R Greathouse IV, Sep 09 2013

Equals A019673/4 or A019675/3 or A019679/2. - Omar E. Pol, Aug 19 2019

Equals (1/10) * Sum_{k>=1} sin(k*Pi/6)/k. - Amiram Eldar, May 30 2021

A197589 Decimal expansion of least x>0 satisfying f(x)=m/2, where m is the maximal value of the function f(x)=cos(x)^2+sin(2x)^2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 18 2011

For a discussion and guide to related sequences, see A197739.

			x=1.12868019433775284470060498453346294726...

Cf. A197739, A195700.

b = 1; c = 2;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .65, .66}, WorkingPrecision -> 110]
RealDigits[r]  (* A195700, arcsin(sqrt(3/8)) *)
m = s[r]
RealDigits[m]
Rationalize[{m, m/2, m/3, 2 m/3, m/4, 3 m/4}]
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197589 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197591 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t] (* A019670, pi/3 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197592 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

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