A248897 -id:A248897 - cp's OEIS frontend

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

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

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

A097109 G.f.: s(2)^2s(3)^3/(s(1)s(6)^2), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

Original entry on oeis.org

1, 1, 0, -2, -3, 0, 0, 2, 0, -2, 0, 0, 6, 2, 0, 0, -3, 0, 0, 2, 0, -4, 0, 0, 0, 1, 0, -2, -6, 0, 0, 2, 0, 0, 0, 0, 6, 2, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, 6, 3, 0, 0, -6, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, -4, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, -2, -6, 0, 0, 2, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 2, 0, 0, -3, 0, 0, 2, 0

Offset: 0

N. J. A. Sloane, Sep 16 2004

Coefficients are multiplicative [Fine].

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.36).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A010815, A115978, A122861, A248897.

QP = QPochhammer; s = QP[q^2]^2*(QP[q^3]^3/(QP[q]*QP[q^6]^2)) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^2), n))} /* Michael Somos, Sep 15 2006 */

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if( p==3, -2, if( p%6==1, e+1, 1-e%2))))))} /* Michael Somos, Sep 15 2006 */

Fine gives an explicit formula for a(n) in terms of the divisors of n.
From Michael Somos, Sep 15 2006: (Start)
Expansion of (a(q) - 3*a(q^3) - 4*a(q^4) + 12*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Euler transform of period 6 sequence [ 1, -1, -2, -1, 1, -2, ...].
a(n) is multiplicative with a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6). (End)
a(3*n + 2) = 0. a(3*n) = A115978(n). a(3*n + 1) = A122861(n).
Sum_{k=0..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 22 2024

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 22 2005

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

Cf. A033687, A033762, A092829, A093829, A097195, A248897, A255648 (absolute values).

QP = QPochhammer; s = QP[q]*QP[q^2]*(QP[q^18]^2/(QP[q^6]*QP[q^9])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1)^e; f[3, e_]:= -2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 28 2024 *)

{a(n)=if(n<1, 0, if(n%3==0, n/=3; -2,1)* sumdiv(n,d,kronecker(-12,d) -if(d%2==0, 2*kronecker(-3,d/2))))}

{a(n)=local(A); if (n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^18+A)^2/ eta(x^6+A)/eta(x^9+A), n))}

Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].
Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].
Multiplicative with a(2^e) = (-1)^e, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} Kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).
a(3n) = -2*A092829(n). a(3n+1) = A093829(3n+1) = A033687(n). a(3n+2) = A093829(3n+2). a(6n)/2 = A093829(n). a(6n+1) = A097195(n). a(6n+3) = -2*A033762(n). a(6n+5) = 0.
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A123331 Expansion of (c(q)^2/(3c(q^2))-1)/2 in powers of q where c(q) is a cubic AGM function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 26 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A123330(n)=2*a(n) if n>0. A113974(n)=-(-1)^n*a(n).

Cf. A248897.

f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (3-(-1)^e)/2; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)

{a(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*kronecker(-3,d)))}

{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, (3-(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}

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

Moebius transform is period 6 sequence [ 1, 1, 0, -1, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (3-(-1)^e)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(3n) = a(4n) = a(n). a(6n+5) = 0.
G.f.: Sum_{k>0} x^k/(1-x^k+x^(2k)) = (theta_3(-q^3)^3/theta_3(-q) - 1)/2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Nov 14 2023

A072703 Indices of Fibonacci numbers whose last digit is 5.

Original entry on oeis.org

5, 10, 20, 25, 35, 40, 50, 55, 65, 70, 80, 85, 95, 100, 110, 115, 125, 130, 140, 145, 155, 160, 170, 175, 185, 190, 200, 205, 215, 220, 230, 235, 245, 250, 260, 265, 275, 280, 290, 295, 305, 310, 320, 325, 335, 340, 350, 355, 365, 370, 380, 385, 395, 400, 410

Offset: 1

Benoit Cloitre, Aug 07 2002

Sequence contains numbers of the forms 5 + 60*k, 10 + 60*k, 20 + 60*k, 25 + 60*k, 35 + 60*k, 40 + 60*k, 50 + 60*k, 55 + 60*k, where k>=0.

Numbers that are congruent to {5, 10} mod 15. - Amiram Eldar, Jan 01 2022, Nov 25 2024

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000045, A001651, A003893, A248897.

Flatten[Position[Fibonacci[Range[400]],?(Last[IntegerDigits[#]]==5&)]] (* or *) LinearRecurrence[{1,1,-1},{5,10,20},60] (* or *) Table[-(5/4) (3+(-1)^n-6 n),{n,60}] (* _Harvey P. Dale, May 15 2011 *)

a(n) = 15*(n-1)-a(n-1), with a(1) = 5. - Vincenzo Librandi, Aug 08 2010
From Harvey P. Dale, May 15 2011: (Start)
a(1) = 5, a(2) = 10, a(3) = 20, a(n) = a(n-1)+a(n-2)-a(n-3).
a(n) = -(5/4)*(3+(-1)^n-6*n). (End)
G.f.: 5*x*(x^2+x+1) / ((x-1)^2*(x+1)). - Colin Barker, Jun 16 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(15*sqrt(3)) = A248897 / 10. - Amiram Eldar, Jan 01 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/10)*sec(Pi/6) = sqrt((5+sqrt(5))/6).
Product_{n>=1} (1 + (-1)^n/a(n)) = (2/sqrt(3))*cos(7*Pi/30). (End)
a(n) = 5 * A001651(n). - Alois P. Heinz, Nov 27 2024

Edited by M. F. Hasler, Oct 08 2014

A138805 Theta series of quadratic form x^2 + xy + 7y^2.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 4, 0, 6, 0, 0, 0, 4, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 6, 4, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 12, 2, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Mar 30 2008

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^7 + 6*q^9 + 4*q^13 + 2*q^16 + 4*q^19 + 2*q^25 + ...

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

Cf. A004016, A138806, A248897.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(27), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[8] + 6*A[10] + 4*A[14] + 2*A[15]; /* Michael Somos, Sep 08 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -3, n/#] {1, 1, 0, 1, 1, 0, 1, 1, 3}[[Mod[#, 9, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^27] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^27], {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 14], n, 1)), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^54 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^27 + A) * eta(x^108 + A))^2 + 4 * x^7 * (eta(x^4 + A) * eta(x^108 + A))^2 / (eta(x^2 + A) * eta(x^54 + A)), n))};

{a(n) = if( n<1, n==0, 2 * sumdiv(n, d, kronecker(-3, n/d) * [ 3, 1, 1, 0, 1, 1, 0, 1, 1][n%9 + 1]))}; /* Michael Somos, Sep 08 2015 */

Expansion of theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) in powers of q.
Expansion of phi(q) * phi(q^27) + 4 * q^7 * psi(q^2) * psi(q^54) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 27 sequence [ 2, -2, -2, 2, -2, 2, 2, -2, 6, 2, -2, -2, 2, -2, 2, 2, -2, -6, 2, -2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b() is multiplicative with b(3^e) = 3 if e>1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{i, j in Z} x^(i*i + i*j + 7*j*j).
a(3*n + 2) = a(4*n + 2) = 0.
a(n) = 2 * A138806(n) unless n=0. a(9*n) = A004016(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 29 2023

A248181 Decimal expansion of Sum_{h >= 0} 1/binomial(h, floor(h/2)).

Original entry on oeis.org

3, 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

Offset: 0

Clark Kimberling, Oct 04 2014

Is this 2 + A248897? [Bruno Berselli, Mar 06 2015]. Yes, see Mathematica program below. - Vaclav Kotesovec, Jul 01 2024

			3.20919957615614523372938550509477048818...
Equals  1 + 1 + 1/2 + 1/3 + 1/6 + 1/10 + 1/20 + 1/35 + 1/70 + 1/126 + ...

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A248182.

r = N[Sum[1/Binomial[h, Floor[h/2]], {h, 0, 2000}], 200];
u = RealDigits[N[r, 200]][[1]]
(* or *)
Sum[1/Binomial[h, h/2], {h, 0, Infinity, 2}] + Sum[1/Binomial[h, (h-1)/2], {h, 1, Infinity, 2}] // Simplify // Expand (* Vaclav Kotesovec, Jul 01 2024 *)

Equals 2 + 2*Pi/3^(3/2). - Vaclav Kotesovec, Jul 01 2024

A255648 Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, May 06 2015

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^6 + 2*q^7 + q^8 + 2*q^9 + 2*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A035178, A248897.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

a[ n_] := If[ n < 1, 0, Sum[ { 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0}[[Mod[ d, 18, 1]]], { d, Divisors[ n]}]];

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1][d%18 + 1]))};

{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, 2, p%6==1, e+1, 1-e%2)))};

Expansion of (b(q^2)^2 / b(q) + b(q^6)^2 / b(q^3) - 2) / 3 in powers of q where b() is a cubic AGM theta function.
Expansion of (psi(q)^3 / psi(q^3) + psi(q^3)^3 / psi(q^9) - 2) / 3 in powers of q where psi() is a Ramanujan theta function.
Moebius transform is period 18 sequence [ 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2 if e>1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k))^2 + (x^(3*k) + x^(9*k)) / (1 + x^(6*k))^2.
a(2*n) = a(n). a(3*n) = 2 * A035178(n). a(3*n + 1) = A033687(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 22 2023

A256923 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)*(1/3)^(2k).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.189958633407180946467791617427446722751559110541442648...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights (2011) p. 272.

Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A073006, A202623, A248897, A256924.

RealDigits[Log[2*Pi/(3*Sqrt[3])], 10, 105] // First

Equals log(Gamma(2/3)*Gamma(4/3)).
Equals log(2*Pi/(3*sqrt(3))).
Equals log(A248897).
Equals -Sum_{k>=1} log(1 - 1/(3*k)^2). - Amiram Eldar, Aug 12 2020

cp's OEIS Frontend

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

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

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A097109 G.f.: s(2)^2*s(3)^3/(s(1)*s(6)^2), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

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A112848 Expansion of eta(q)*eta(q^2)*eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

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A123331 Expansion of (c(q)^2/(3c(q^2))-1)/2 in powers of q where c(q) is a cubic AGM function.

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A072703 Indices of Fibonacci numbers whose last digit is 5.

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A138805 Theta series of quadratic form x^2 + x*y + 7*y^2.

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

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

A097109 G.f.: s(2)^2s(3)^3/(s(1)s(6)^2), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

A138805 Theta series of quadratic form x^2 + xy + 7y^2.