A073010 -id:A073010 - cp's OEIS frontend

A138806 Expansion of (theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) - 1) / 2 in powers of q.

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

Offset: 1

Michael Somos, Mar 30 2008

Half the number of integer solutions to x^2 + x*y + 7*y^2 = n. - Jianing Song, Nov 20 2019

			q + q^4 + 2*q^7 + 3*q^9 + 2*q^13 + q^16 + 2*q^19 + q^25 + 3*q^27 + ...

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A138805 (number of integer solutions to x^2 + x*y + 7*y^2 = n).
Cf. A033687, A073010, A097195, A123884, A121361.
Similar sequences: A096936, A113406, A110399.

f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := 1 - Mod[e, 2]; f[3, e_] := 3; f[3, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

{a(n) = if( n<1, 0, if( n%3 == 2, 0, if( n%3==1, sumdiv(n, d, kronecker(-3, d)), if( n%9==0, 3 * sumdiv(n/9, d, kronecker(-3, d))))))}

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-3, d)) - if( n%3==0, sumdiv(n/3, d, [0, 1, -1, -3, 1, -1, 3, 1, -1][d%9+1])))}

{a(n) = if( n<1, 0, qfrep([2, 1; 1, 14], n, 1)[n])}

a(n) is multiplicative and a(3^e) = 3 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).
a(3*n + 2) = a(4*n + 2) = 0.
G.f.: (Sum_{i,j} x^(i*i + i*j + 7*j*j) - 1) / 2.
A138805(n) = 2 * a(n) unless n=0. A033687(n) = a(3*n + 1). A097195(n) = a(6*n + 1). A123884(n) = a(12*n + 1). 2 * A121361(n) = a(12*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 16 2023

A166517 a(n) = (3 + 5(-1)^n + 6n)/4.

Original entry on oeis.org

2, 1, 5, 4, 8, 7, 11, 10, 14, 13, 17, 16, 20, 19, 23, 22, 26, 25, 29, 28, 32, 31, 35, 34, 38, 37, 41, 40, 44, 43, 47, 46, 50, 49, 53, 52, 56, 55, 59, 58, 62, 61, 65, 64, 68, 67, 71, 70, 74, 73, 77, 76, 80, 79, 83, 82, 86, 85, 89, 88, 92, 91, 95, 94, 98, 97, 101, 100, 104, 103, 107

Offset: 0

Vincenzo Librandi, Oct 16 2009

A sequence defined by a(1)=1, a(n)=k*n-a(n-1), k a constant parameter, has recurrence a(n)= 3*a(n-1) -3*a(n-2) +a(n-3). Its generating function is x*(1+2*(k-1)*x+(1-k)*x^2)/((1+x)*(1-x)^2). The closed form is a(n) = k*n/2+k/4+(-1)^n*(3*k/4-1). This applies with k=3 to this sequence here, and for example to sequences A165033, and A166519-A166525. - R. J. Mathar, Oct 17 2009
From Paul Curtz, Feb 20 2010: (Start)
Also: A001651, terms swapped by pairs.
a(n) mod 9 defines a period-6 sequence which is a permutation of A141425. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A010685, A010716, A016777, A016945, A073010.

[(3 +5*(-1)^n+6*n)/4: n in [0..80]]; // Vincenzo Librandi, Sep 13 2013

CoefficientList[Series[(2 x^2 - x + 2)/((1 + x) (x - 1)^2), {x, 0, 80}], x] (* Harvey P. Dale, Mar 25 2011 *)
Table[(3 + 5 (-1)^n + 6 n) / 4, {n, 0, 100}] (* Vincenzo Librandi, Sep 13 2013 *)

a(n) = 3*n - a(n-1).
From Paul Curtz, Feb 20 2010: (Start)
a(n+1)-a(n) = (-1)^(n+1)*A010685(n).
Second differences: |a(n+2)-2*a(n+1)+a(n)| = A010716(n).
a(2*n) + a(2*n+1) = A016945(n) = 6*n+3.
a(2*n) = A016945(n).
a(2*n+1) = A016777(n). (End)
G.f. ( 2-x+2*x^2 ) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Mar 08 2011
E.g.f.: (1/4)*exp(-x)*(5 + 3*exp(2*x) + 6*x*exp(2*x)). - G. C. Greubel, May 15 2016
Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 24 2023

a(0)=2 added by Paul Curtz, Feb 20 2010

A260658 Numerators of a BBP-like formula for 4*Pi/sqrt(27).

Original entry on oeis.org

5, -7, 23, -1, 41, -25, 59, -17, 77, -43, 95, -13, 113, -61, 131, -35, 149, -79, 167, -11, 185, -97, 203, -53, 221, -115, 239, -31, 257, -133, 275, -71, 293, -151, 311, -5, 329, -169, 347, -89, 365, -187, 383, -49, 401, -205, 419, -107, 437, -223, 455, -29, 473

Offset: 0

David Brink, Nov 13 2015

4*Pi/sqrt(27) = Sum_{n >= 0} (-1/8)^n*(2/(3*n+1)+1/(3*n+2)).

The reduced Collatz function R applied to the numbers 6n+3: a(n) = R(6n+3), where R(k) = (3k+1)/2^r, with r as large as possible, yields an unsigned version of this sequence. - Jonas Kaiser, Jun 17 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants.
David Brink, Nilakantha's accelerated series for pi, Acta Arith. 171 (2015), 293-308.

Cf. A073010, A260659 (denominators).

[Numerator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))): n in [0..60]]; // Vincenzo Librandi, Nov 20 2015

A260658[n_] := Numerator[(-1/8)^n*(2/(3*n + 1) + 1/(3*n + 2))];
Array[A260658, 100, 0] (* Paolo Xausa, Jun 19 2024 *)

a(n) = numerator((-1/8)^n*(2/(3*n+1) + 1/(3*n+2))); \\ Michel Marcus, Nov 15 2015

a(n) = numerator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))).

More terms from Michel Marcus, Nov 15 2015

A293899 Number of proper divisors of the form 3k+1 minus number of proper divisors of the form 3k+2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 06 2017

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002324, A010872, A073010, A293895, A293896.

Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 1, # != n] &] - DivisorSum[n, 1 &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
Table[Total[Which[Mod[#,3]==1,1,Mod[#,3]==2,-1,True,0]&/@Most[ Divisors[ n]]],{n,110}] (* Harvey P. Dale, Nov 26 2021 *)

A293895(n) = sumdiv(n,d,(dA293896(n) = sumdiv(n,d,(dA293899(n) = (A293895(n) - A293896(n));

When n = 3k, a(n) = A002324(n), when n = 3k+1, a(n) = A002324(n) - 1, when n = 3k+2, a(n) = A002324(n) + 1.
a(n) = A002324(n) - A010872(n) (mod 3).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 25 2023

A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

Imaginary part associated with A204067.

			0.446489755784624605609282...

R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A073005, A073010, A073006, A202623, A204067

Maple
```
evalf(Pi/GAMMA(2/3)/3^(3/2) ) ;
```

RealDigits[Gamma[1/3]/6, 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Equals Pi/(Gamma(2/3)* 3^(3/2)) = A073010 / A073006.
(this value)^2 + A204067^2 = A202623^2.
Equals Gamma(1/3)/6 = A073005 / 6. - Amiram Eldar, May 26 2023

A261839 Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.5054294746835192416424504819084321491886690145682628649826647...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)).

S[5] = 2*Pi*Im[PolyLog[4, (-1)^(1/3)]] + (1/9)*Pi^2*Zeta[3] - 19*Zeta[5]/3; RealDigits[S[5], 10, 103] // First

suminf(n=1, 1/(n^5*binomial(2*n,n))) \\ Michel Marcus, Sep 03 2015

S(5) = 2*Pi*Im(PolyLog(4, (-1)^(1/3))) + (1/9)*Pi^2*zeta(3) -19*zeta(5)/3.
Equals (1/2) 4F3(1,1,1,1; 3/2,2,2; 1/4).
Also equals (1/(2592*sqrt(3)))*(Pi*(PolyGamma(3, 1/6) + PolyGamma(3, 1/3) - PolyGamma(3, 2/3) - PolyGamma(3, 5/6))) + (1/9)*Pi^2*zeta(3) - 19*zeta(5)/3.

A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 19 2019

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A175629 and s = 1.

			1 + 1/2 - 1/3 + 1/4 - 1/5 - 1/6 + 1/8 + 1/9 - 1/10 + 1/11 - 1/12 - 1/13 + ... = Pi/sqrt(7) = 1.1874104117...

Sergio A. Carrillo, Where did the examples of Abel's continuity theorem go?, arXiv:2010.10290 [math.HO], 2020. See p. 11.
Eric Weisstein's World of Mathematics, Class Number.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A175629.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k, where d is a fundamental discriminant: A093954 (d=-8), this sequence (d=-7), A003881 (d=-4), A073010 (d=-3), A086466 (d=5), A196525 (d=8), A196530 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: this sequence (s=1), A103133 (s=2), A327135 (s=3).

RealDigits[Pi/Sqrt[7], 10, 102] // First

PARI
```
default(realprecision, 100); Pi/sqrt(7)
```

Equals Pi/sqrt(7). This is related to the class number formula: if d<0 is the fundamental discriminant of an imaginary quadratic number field, Chi(k) = Kronecker(d,k), then L(1,Chi) = Sum_{k>=1} Kronecker(d,k)/k = 2*Pi*h(d)/(sqrt(|d|)*w(d)), where h(d) is the class number of K = Q[sqrt(d)], w(d) is the number of elements in K whose norms are 1 (w(d) = 6 if d = -3, 4 if d = -4 and 2 if d < -4). Here d = -7, h(d) = 1, w(d) = 2.
Equals (polylog(1,u) + polylog(1,u^2) - polylog(1,u^3) + polylog(1,u^4) - polylog(1,u^5) - polylog(1,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).
Equals (polygamma(0,1/7) + polygamma(0,2/7) - polygamma(0,3/7) + polygamma(0,4/7) - polygamma(0,5/7) - polygamma(0,6/7))/49.
Equals 1/Product_{p prime} (1 - Kronecker(-7,p)/p), where Kronecker(-7,p) = 0 if p = 7, 1 if p == 1, 2 or 4 (mod 7) or -1 if p == 3, 5 or 6 (mod 7). - Amiram Eldar, Dec 17 2023

A087694 Number of solutions to x^2 + xy + y^2 == 0 (mod n).

Original entry on oeis.org

1, 1, 3, 4, 1, 3, 13, 4, 9, 1, 1, 12, 25, 13, 3, 16, 1, 9, 37, 4, 39, 1, 1, 12, 25, 25, 27, 52, 1, 3, 61, 16, 3, 1, 13, 36, 73, 37, 75, 4, 1, 39, 85, 4, 9, 1, 1, 48, 133, 25, 3, 100, 1, 27, 1, 52, 111, 1, 1, 12, 121, 61, 117, 64, 25, 3, 133, 4, 3, 13

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000086, A073010, A086724.

A087694 := proc(n) option remember; local pf,p,f,e ; if n = 1 then 1; else pf := ifactors(n)[2] ; if nops(pf) = 1 then f := op(1,pf) ; p := op(1,f) ; e := op(2,f) ; if p = 3 then n ; elif p mod 3 =1 then ((p-1)*e+p)*p^(e-1) ; else p^(2*floor(e/2)) ; end if; else mul(procname(op(1,p)^op(2,p)),p=pf) ; end if; end if; end proc:
seq(A087694(n),n=1..70) ; # R. J. Mathar, Jan 07 2011

a[n_] := If[n==1, 1, Product[{p, e} = pe; Which[p==3, 3^e, Mod[p, 3] == 2, (p^2)^Quotient[e, 2], True, ((p-1) e + p) p^(e-1)], {pe, FactorInteger[n] }]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==3, 3^e, if(p%3==2, (p^2)^(e\2), ((p-1)*e+p)*p^(e-1))))} \\ Andrew Howroyd, Jul 09 2018

Multiplicative with a(3^e) = 3^e, a(p^e) = ((p-1)*e+p)*p^(e-1) if p mod 3 = 1, a(p^e) = p^(2*floor(e/2)) if p mod 3 = 2. - Vladeta Jovovic, Sep 27 2003

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A073010/A086724 = 0.77383581325017004332... . - Amiram Eldar, Nov 21 2023

A225847 Decimal expansion of Sum_{n>=1} 1/(nbinomial(4n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 17 2013

			0.269523929027742017317181647486329302840849825343266309815843772918628369827...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 60.

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp. 220 (2013) 331-338, Example 7.

Cf. A073010, A210453, A378802.

(1/4)*HypergeometricPFQ[{1, 1, 4/3, 5/3}, {5/4, 3/2, 7/4}, 27/256] // RealDigits[#, 10, 105]& // First

Equals Integral_{x>0} ((3*x)/((1 + x)*(1 + 3*x + 6*x^2 + 4*x^3 + x^4))) dx.

Equals (3*c/(2*c^2+1)) * log((c-1)/(c+1)) + (3*(c-1)/(2*(2*c^2+1))) * sqrt(c/(c+2)) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1)) + (3*(c+1)/(2*(2*c^2+1))) * sqrt(c/(c-2)) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1)), where c = sqrt(1 + (16/sqrt(3)) * cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013). - Amiram Eldar, Dec 07 2024

A260659 Denominators of a BBP-like formula for 4*Pi/sqrt(27).

Original entry on oeis.org

2, 80, 3584, 1760, 745472, 4456448, 99614720, 265289728, 10905190400, 54492397568, 1065151889408, 1277752770560, 96619584290816, 450799767388160, 8321103999008768, 19017153114013696, 689613692941107200, 3102980143258271744, 55484347409204510720, 30822635849723674624

Offset: 0

David Brink, Nov 13 2015

4*Pi/sqrt(27) = Sum_{n >= 0} (-1/8)^n*(2/(3*n+1)+1/(3*n+2)).

Paolo Xausa, Table of n, a(n) for n = 0..1000
David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants.
David Brink, Nilakantha's accelerated series for pi, Acta Arith. 171 (2015), 293-308.

Cf. A073010, A260658 (numerators).

[Denominator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))): n in [0..60]]; // Vincenzo Librandi, Nov 20 2015

A260659[n_] := Denominator[(-1/8)^n*(2/(3*n + 1) + 1/(3*n + 2))];
Array[A260659, 25, 0] (* Paolo Xausa, Jun 19 2024 *)

a(n) = denominator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))); \\ Michel Marcus, Nov 15 2015

a(n) = denominator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))).

More terms from Michel Marcus, Nov 15 2015

cp's OEIS Frontend

A138806 Expansion of (theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) - 1) / 2 in powers of q.

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A166517 a(n) = (3 + 5*(-1)^n + 6*n)/4.

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A260658 Numerators of a BBP-like formula for 4*Pi/sqrt(27).

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A293899 Number of proper divisors of the form 3k+1 minus number of proper divisors of the form 3k+2.

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A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

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A261839 Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).

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A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

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A166517 a(n) = (3 + 5(-1)^n + 6n)/4.

A225847 Decimal expansion of Sum_{n>=1} 1/(nbinomial(4n,n)).