A214549 -id:A214549 - cp's OEIS frontend

A011655 Period 3: repeat [0, 1, 1].

0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Offset: 0

N. J. A. Sloane

A binary m-sequence: expansion of reciprocal of x^2+x+1 (mod 2).
A Chebyshev transform of the Jacobsthal numbers A001045: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Feb 16 2004
This is the r = 1 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.
This is the Fibonacci sequence (A000045) modulo 2. - Stephen Jordan (sjordan(AT)mit.edu), Sep 10 2007
For n > 0: a(n) = A084937(n-1) mod 2. - Reinhard Zumkeller, Dec 16 2007
This is also the Lucas numbers (A000032) mod 2. In general, this is the parity of any Lucas sequence associated with any pair (P,Q) when P and Q are odd; i.e., a(n) = U_n(P,Q) mod 2 = V_n(P,Q) mod 2. See Ribenboim. - Rick L. Shepherd, Feb 07 2009
Starting with offset 1: (1, 1, 0, 1, 1, 0, ...) = INVERTi transform of the tribonacci sequence A001590 starting (1, 2, 3, 6, 11, 20, 37, ...). - Gary W. Adamson, May 04 2009
From Reinhard Zumkeller, Nov 30 2009: (Start)
Characteristic function of numbers coprime to 3.
a(n) = 1 - A079978(n); a(A001651(n)) = 1; a(A008585(n)) = 0;
A000212(n) = Sum_{k=0..n} a(k)*(n-k). (End)
Sum_{k>0} a(k)/k/2^k = log(7)/3. - Jaume Oliver Lafont, Jun 01 2010
The sequence is the principal Dirichlet character of the reduced residue system mod 3 (the other is A102283). Associated Dirichlet L-functions are L(2,chi) = Sum_{n>=1} a(n)/n^2 = 4*Pi^2/27 = A214549, and L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.157536... = -(psi''(1/3) + psi''(2/3))/54 where psi'' is the tetragamma function. [Jolley eq 309 and arXiv:1008.2547, L(m = 3, r = 1, s)]. - R. J. Mathar, Jul 15 2010
a(n+1), n >= 0, is the sequence of the row sums of the Riordan triangle A158454. - Wolfdieter Lang, Dec 18 2010
Removing the first two elements and keeping the offset at 0, this is a periodic sequence (1, 0, 1, 1, 0, 1, ...). Its INVERTi transform is (1, -1, 2, -2, 2, -2, ...) with period (2,-2). - Gary W. Adamson, Jan 21 2011
Column k = 1 of triangle in A198295. - Philippe Deléham, Jan 31 2012
The set of natural numbers, A000027: (1, 2, 3, ...); is the INVERT transform of the signed periodic sequence (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013
Any integer sequence s(n) = |s(n-1) - s(n-2)| (equivalently, max(s(n-1), s(n-2)) - min(s(n-1), s(n-2))) for n > i + 1 with s(i) = j and s(i+1) = k, where j and k are not both 0, is or eventually becomes a multiple of this sequence, namely, the sequence repeat gcd(j, k), gcd(j, k), 0 (at some offset). In particular, if j and k are coprime, then s(n) is or eventually becomes this sequence (see, e.g., A110044). - Rick L. Shepherd, Jan 21 2014
For n >= 1, a(n) is also the characteristic function for rational g-adic integers (+n/3)A001651).%20See%20the%20definition%20in%20the%20Mahler%20reference,%20p.%207%20and%20also%20p.%2010.%20-%20_Wolfdieter%20Lang">g and also (-n/3)_g for all integers g >= 2 without a factor 3 (A001651). See the definition in the Mahler reference, p. 7 and also p. 10. - _Wolfdieter Lang, Jul 11 2014
Characteristic function for A007908(n+1) being divisible by 3. a(n) = bit flipped A007908(n+1) (mod 3) = bit flipped A079978(n). - Wolfdieter Lang, Jun 12 2017
Also Jacobi or Kronecker symbol (n/9) (or (n/9^e) for all e >= 1). - Jianing Song, Jul 09 2018
The binomial trans. is 0, 1, 3, 6, 11, 21, 42, 85, 171, 342,.. (see A024495). - R. J. Mathar, Feb 25 2023

			G.f. = x + x^2 + x^4 + x^5 + x^7 + x^8 + x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + ...

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
H. D. Lueke, Korrelationssignale, Springer 1992, pp. 43-48.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.
K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.
Paulo Ribenboim, The Little Book of Big Primes. Springer-Verlag, NY, 1991, p. 46. [Rick L. Shepherd, Feb 07 2009]

Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
L. B. W. Jolley, Summation of Series, Dover, (1961).
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,1).
Index entries for characteristic functions.
Index entries for sequences related to Chebyshev polynomials..

Partial sums of A057078 give A011655(n+1).
Cf. A035191 (Mobius transform), A001590, A002487, A049347.
Cf. A000035, A011558, A097325, A109720, A145568, A166486, A168181, A168182, A168184, A168185. - Reinhard Zumkeller, Nov 30 2009
Cf. A000027, A000045, A004523 (partial sums), A057078 (first differences).
Cf. A007908, A079978 (bit flipped).
Cf. A011656 - A011751 for other binary m-sequences.
Cf. A002264.

a011655 = fromEnum . ((/= 0) . (`mod` 3))
a011655_list = cycle [0,1,1]  -- Reinhard Zumkeller, Apr 07 2012

[(n^2 mod 3) : n in [0..100]]; // Wesley Ivan Hurt, Apr 16 2015

A011655:=n->(n^2 mod 3): seq(A011655(n), n=0..100); # Wesley Ivan Hurt, Apr 16 2015

A011655[n_] := If[Mod[n, 3] == 0, 0, 1];  Array[A011655, 105, 0] (* Robert G. Wilson v *)
Mod[Fibonacci[Range[0, 99]], 2] (* Alonso del Arte, Jul 20 2017 *)

PARI
```
{a(n) = sign(n%3)};
```

a(n)=!!(n%3) \\ Jaume Oliver Lafont, Mar 24 2009

a(n)=n%3>0 \\ M. F. Hasler, Feb 17 2018

def A011655(n): return int(bool(n%3)) # Chai Wah Wu, May 25 2022

G.f.: (x + x^2) / (1 - x^3) = Sum_{k>0} (x^k - x^(3*k)).
G.f.: x / (1 - x / (1 + x / (1 + x / (1 - 2*x / (1 + x))))). - Michael Somos, Apr 02 2012
a(n) = a(n+3) = a(-n), a(3*n) = 0, a(3*n + 1) = a(3*n + 2) = 1 for all n in Z.
a(n) = (1/2)*( (-1)^(floor((2n + 4)/3)) + 1 ). - Mario Catalani (mario.catalani(AT)unito.it), Oct 22 2003
a(n) = Fibonacci(n) mod 2. - Paul Barry, Nov 12 2003
a(n) = (2/3)*(1 - cos(2*Pi*n/3)). - Ralf Stephan, Jan 06 2004
a(n) = 1 - a(n-1)*a(n-2), a(n) = n for n < 2. - Reinhard Zumkeller, Feb 28 2004
a(n) = 2*(1 - T(n, -1/2))/3 with Chebyshev's polynomials T(n, x) of the first kind; see A053120. - Wolfdieter Lang, Oct 18 2004
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*A001045(n-2k)/(n-k). - Paul Barry, Oct 31 2004
a(n) = A002487(n) mod 2. - Paul Barry, Jan 14 2005
From Bruce Corrigan (scentman(AT)myfamily.com), Aug 08 2005: (Start)
a(n) = n^2 mod 3.
a(n) = (1/3)*(2 - (r^n + r^(2*n))) where r = (-1 + sqrt(-3))/2. (End)
From Michael Somos, Sep 23 2005: (Start)
Euler transform of length 3 sequence [ 1, -1, 1].
Moebius transform is length 3 sequence [ 1, 0, -1].
Multiplicative with a(3^e) = 0^e, a(p^e) = 1 otherwise. (End)
From Hieronymus Fischer, Jun 27 2007: (Start)
a(n) = (4/3)*(|sin(Pi*(n-2)/3)| + |sin(Pi*(n-1)/3)|)*|sin(Pi*n/3)|.
a(n) = ((n+1) mod 3 + 1) mod 2 = (1 - (-1)^(n - 3*floor((n+1)/3)))/2. (End)
a(n) = 2 - a(n-1) - a(n-2) for n > 1. - Reinhard Zumkeller, Apr 13 2008
a(2*n+1) = a(n+1) XOR a(n), a(2*n) = a(n), a(1) = 1, a(0) = 0. - Reinhard Zumkeller, Dec 27 2008
Sum_{n>=1} a(n)/n^s = (1-1/3^s)*Riemann_zeta(s), s > 1. - R. J. Mathar, Jul 31 2010
a(n) = floor((4*n-5)/3) mod 2. - Gary Detlefs, May 15 2011
a(n) = (a(n-1) - a(n-2))^2 with a(0) = 0, a(1) = 1. - Francesco Daddi, Aug 02 2011
Convolution of A040000 with A049347. - R. J. Mathar, Jul 21 2012
G.f.: Sum_{k>0} x^A001651(k). - L. Edson Jeffery, Dec 05 2012
G.f.: x/(G(0) - x^2) where G(k) = 1 - x/(x + 1/(1 - x/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 15 2013
For the general case: The characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, with m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = sign(n mod 3). - Wesley Ivan Hurt, Jun 22 2013
a(n) = A000035(A000032(n)) = A000035(A000045(n)). - Omar E. Pol, Oct 28 2013
a(n) = (-n mod 3)^((n-1) mod 3). - Wesley Ivan Hurt, Apr 16 2015
a(n) = (2/3) * (1 - sin((Pi/6) * (4*n + 3))) for n >= 0. - Werner Schulte, Jul 20 2017
a(n) = a(n-1) XOR a(n-2) with a(0) = 0, a(1) = 1. - Chunqing Liu, Dec 18 2022
a(n) = floor((n+2)/3) - floor(n/3) = A002264(n+2) - A002264(n). - Aaron J Grech, Jul 30 2024
E.g.f.: 2*(exp(x) - exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Mar 30 2025
Dirichlet g.f.: zeta(s)*(1-1/3^s). - R. J. Mathar, Aug 10 2025

Better name from Omar E. Pol, Oct 28 2013

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

Original entry on oeis.org

1, 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, 8

Offset: 1

Bruno Berselli, Mar 06 2015

Value of the Borwein-Borwein function I_3(a,b) for a = b = 1. - Stanislav Sykora, Apr 16 2015

The area of a circle circumscribing a unit-area regular hexagon. - Amiram Eldar, Nov 05 2020

			1.2091995761561452337293855050947704881893774987284937170465899569254...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), pp. 120-121.
L. B. W. Jolley, Summation of Series, Dover (1961), No. 261, pp. 48, 49, (and No. 275).

Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).
Su Hu and Min-Soo Kim, A generalization of Wallis' formula, arXiv:2201.09674 [math.NT], 2022.
Richard Kershner, The Number of Circles Covering a Set, American Journal of Mathematics, 61(3), 665-671.
MIT Integration Bee, 2023 MIT Integration Bee - Finals, Problem 1.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), C6.2.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See (9) p. 146.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean, equations 26-32.
Index entries for transcendental numbers

Cf. A000796, A001651, A002194, A093602, A248181, A257096, A257097, A186706.
Cf. A010815, A073010, A086089, A214549, A256923, A275486.
Cf. A091682 (Sum_{i >= 0} (i!)^2/(2*i)!).

RealDigits[2 Sqrt[3] Pi/9, 10, 100][[1]]

a = 2*Pi/(3*sqrt(3)) \\ Stanislav Sykora, Apr 16 2015

Equals 2*sqrt(3)*Pi/9 = 1 + 1/6 + 1/30 + 1/140 + 1/630 + 1/2772 + 1/12012 + ...
Equals m*I_3(m,m) = m*Integral_{x>=0} (x/(m^3+x^3)), for any m>0. - Stanislav Sykora, Apr 16 2015
Equals Integral_{x>=0} (1/(1+x^3)) dx. - Robert FERREOL, Dec 23 2016
From Peter Bala, Oct 27 2019: (Start)
Equals 3/4*Sum_{n >= 0} (n+1)!*(n+2)!/(2*n+3)!.
Equals Sum_{n >= 1} 3^(n-1)/(n*binomial(2*n,n)).
Equals 2*Sum_{n >= 1} 1/(n*binomial(2*n,n)). See Boros and Moll, pp. 120-121.
Equals Integral_{x = 0..1} 1/(1 - x^3)^(1/3) dx = Sum_{n >= 0} (-1)^n*binomial(-1/3,n) /(3*n + 1).
Equals 2*Sum_{n >= 1} 1/((3*n-1)*(3*n-2)) = 2*(1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...) (added Oct 30 2019). (End)
Equals Product_{k>=1} 9*k^2/(9*k^2 - 1). - Amiram Eldar, Aug 04 2020
From Peter Bala, Dec 13 2021: (Start)
Equals (2/3)*A093602.
Conjecture: for k >= 0, 2*sqrt(3)*Pi/9 = (3/2)^k * k!*Sum_{n = -oo..oo} (-1)^n/ Product_{j = 0..k} (3*n + 3*j + 1). (End)
Equals (3/4)*S - 1, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..Pi/2} tan(x)^(1/3)/(sin(2*x) + 1) dx. See MIT Link. - Joost de Winter, Aug 26 2023
Continued fraction: 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). See A000407. - Peter Bala, Feb 20 2024
Equals Sum_{n>=2} 1/binomial(n, floor(n/2)); and trivially if "floor" is replaced by "ceiling". - Richard R. Forberg, Aug 30 2024
Equals Product_{k>=2} (1 + (-1)^k/A001651(k)). - Amiram Eldar, Nov 22 2024
Equals 2*A073010 = 1/A086089 = sqrt(A214549) = exp(A256923) = A275486/2. - Hugo Pfoertner, Nov 22 2024
Equals 1 - (1/6) * Sum_{n>=1} A010815(n)/n. - Friedjof Tellkamp, Apr 05 2025
Equals A248181 - 2. - Pontus von Brömssen, Apr 05 2025

A333240 Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Peter Luschny, May 13 2020

The range of product are the primes of the form 3*k - 1 (A003627).

See a comment of R. J. Mathar in A175646.

			1.414064390892147637565501819079829379907695069393162175039924962423928106992...

Peter Luschny, Table of n, a(n) for n = 1..1000
Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, c_3 in formula (1.8) and (5.6), alternative link.
S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
Jerzy Kaczorowski, Waldemar Ratajczak, Peter Nijkamp, Krzysztof Górnisiewicz, Economic hierarchical spatial systems - new properties of Löschian numbers, Applied Mathematics and Computation, Volume 461 (2024) 128319. (The product appears in Theorem 3 (12)).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{3,2}(2) in section 3.2.

Cf. A003627, A004612, A175646, A214549, A301429, A333239.

z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:
evalf(mul(x(n), n=1..8), 105); # Peter Luschny, Jan 17 2021

digits = 104; precision = digits + 10;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;
RealDigits[pgv, 10, digits][[1]]
(* Jean-François Alcover, Jan 12 2021, after PARI code due to Artur Jasinski *)
z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;
N[Product[x[n], {n, 8}], 105] (* Peter Luschny, Jan 17 2021 *)

A333240 * A175646 = (4*Pi^2)/27 = A214549.
A301429 = sqrt(A333240) / 12^(1/4).
Equals Sum_{k>=1} 1/A004612(k)^2. - Amiram Eldar, Sep 27 2020

Last 5 digits corrected by Jean-François Alcover, Jan 12 2021

Better name by Peter Luschny, Jan 17 2021

A086724 Decimal expansion of L(2, chi3) = g(1)-g(2)+g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

This number is L(2, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3, A102283. - Stuart Clary, Dec 17 2008
Equals 1/1^2 -1/2^2 +1/4^2 -1/5^2 +1/7^2 -1/8^2 +1/10^2 -1/11^2 +-... . This can be split as (1/1^2 -1/5^2 +1/7^2 -1/11^2 +-...) - (1/2^2 -1/4^2 +1/8^2 -1/10^2..) = (g(1)-g(5)) - (g(2)-g(4)). The first of these two series is A214552 and the second series is 1/(2^2)*(1-1/2^2 +1/4^2-1/5^2+-...), namely a quarter of the original series. Therefore 5/4 of this value here equals A214552. - R. J. Mathar, Jul 20 2012
Calegari, Dimitrov, & Tang prove that this number is irrational. - Charles R Greathouse IV, Aug 29 2024

			0.781302412896486296867...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Integrals of the Ising class, J. Phys. A 39 (2006) 12271, variable C_3.
Frank Calegari, Vesselin Dimitrov, and Yunqing Tang, The linear independence of 1, ζ(2), and L(2,χ₋₃), arXiv:2408.15403 [math.NT], 2024.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98.
Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=3,r=2,s=2).
Lorenz Milla, World record computation of Dirichlet L(-3,2) series (1,000,000,000,000 digits)
Kh. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, El. J. Combin. 18 (2) (2012), Article P35, value of K after Theorem 4.

Cf. A086722-A086731, A102283, A214549 (principal character), A214552.

Cf. A153066, A153067, A153068.

nmax = 1000; First[ RealDigits[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, 10, nmax] ] (* Stuart Clary, Dec 17 2008 *)

zetahurwitz(2,1/3)/9 - zetahurwitz(2,2/3)/9 \\ Charles R Greathouse IV, Jan 30 2018

From Jean-François Alcover, Jul 17 2014, updated Jan 23 2015: (Start)
Equals Sum_{n>=1} jacobi(-3, n+3)/n^2.
Equals (8/15)*4F3(1/2,1,1,2; 5/4,3/2,7/4; 3/4), where 4F3 is the generalized hypergeometric function.
Equals 4*Pi*log(3)/(3*sqrt(3)) - 4*Integral_{0..1} log(x+1)/(x^2-x+1) dx. (End)
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^2)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^2)^(-1). - Amiram Eldar, Nov 06 2023

A214550 Decimal expansion of Sum_{n>=0} 1/(3*n+1)^2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jul 20 2012

Sum over the inverse squares of A016777. Dirichlet series Sum_{n>=1} A079978(n-1)/n^s at s=2.

This is also (1/9)*Zeta(2, 1/3) = (1/9)*Psi(1, 1/3) with the Hurwitz zeta function Zeta(s, a) and the Polygamma function Psi(n, z). See the programs. - Wolfdieter Lang, Nov 12 2017

			1.1217330139363437868657... = 1/1^2 + 1/4^2 + 1/7^2 + 1/10^2 + 1/13^2 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A016777, A086724, A214549, A294967.

Maple
```
evalf(Psi(1,1/3)/9);
```

RealDigits[ PolyGamma[1, 1/3]/9, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

zetahurwitz(2,1/3)/9 \\ Charles R Greathouse IV, Jan 30 2018

sumpos(n=0,1/(3*n+1)^2) \\ Charles R Greathouse IV, Jan 30 2018

Equals (A086724 + A214549)/2 because the sequence represented by A079978 (with offset 1) is the average of A011655 and A102283.
From Amiram Eldar, Oct 02 2020: (Start)
Equals Integral_{0..1} log(x)/(x^3-1) dx = Integral_{1..oo} x*log(x)/(x^3-1) dx.
Equals 4*Pi^2/27 - A294967. (End)

More terms from Jean-François Alcover, Feb 11 2013

A135556 Squares of numbers not divisible by 3: a(n) = A001651(n)^2.

Original entry on oeis.org

1, 4, 16, 25, 49, 64, 100, 121, 169, 196, 256, 289, 361, 400, 484, 529, 625, 676, 784, 841, 961, 1024, 1156, 1225, 1369, 1444, 1600, 1681, 1849, 1936, 2116, 2209, 2401, 2500, 2704, 2809, 3025, 3136, 3364, 3481, 3721, 3844, 4096, 4225, 4489, 4624, 4900

Offset: 1

Artur Jasinski, Nov 25 2007

From Fermat's Little Theorem all these numbers are congruent to 1 mod 3.
From Peter Bala, Jan 26 2025: (Start)
The sequence terms are the exponents of q in the expansion of q*Product_{n >= 1} (1 - q^(3*n))*(1 - q^(18*n))^2/( (1 - q^(6*n))*(1 - q^(9*n)) ) = q - q^4 - q^16 + q^25 + q^49 - q^64 - q^100 + + - - ....
Also, the exponents of q in the expansion of q*Product_{n >= 1} (1 - q^(6*n))^5/(1 - q^(3*n))^2 = q + 2*q^4 - 4*q^16 - 5*q^25 + 7*q^49 + 8*q^64 -10*q^100 - 11*q^121 + + - - ....  See Lemke Oliver, Theorem 1.2. (End)

Colin Barker, Table of n, a(n) for n = 1..1000
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651, A001082, A214549.

Partial sums of A298028.

LinearRecurrence[{1,2,-2,-1,1}, {1,4,16,25,49}, 25] (* or *) Table[(18*n^2-6*(-1)^n*n-18*n+3*(-1)^n+5)/8, {n,1,25}] (* G. C. Greubel, Oct 19 2016 *)
Flatten[Partition[Range[70],2,3,{1,1},{}]]^2 (* Harvey P. Dale, Jun 19 2018 *)

isok(n) = issquare(n) && (n % 3 == 1); \\ Michel Marcus, Nov 02 2013

Vec(-x*(1+3*x+10*x^2+3*x^3+x^4) / ( (1+x)^2*(x-1)^3 ) + O(x^100)) \\ Colin Barker, Jan 26 2016

G.f.: -x*(1+3*x+10*x^2+3*x^3+x^4) / ((1+x)^2*(x-1)^3). - R. J. Mathar, Feb 16 2011
From Colin Barker, Jan 26 2016: (Start)
a(n) = (18*n^2-6*(-1)^n*n-18*n+3*(-1)^n+5)/8.
a(n) = (9*n^2-12*n+4)/4 for n even.
a(n) = (9*n^2-6*n+1)/4 for n odd. (End)
E.g.f.: (1/8)*( (3 + 6*x)*exp(-x) - 8 + (5 + 18*x^2)*exp(x)). - G. C. Greubel, Oct 19 2016
Sum_{n>=1} 1/a(n) = 4*Pi^2/27 (A214549). - Amiram Eldar, Dec 19 2020

cp's OEIS Frontend

A011655 Period 3: repeat [0, 1, 1].

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A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

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A333240 Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2).

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A086724 Decimal expansion of L(2, chi3) = g(1)-g(2)+g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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A214550 Decimal expansion of Sum_{n>=0} 1/(3*n+1)^2.

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A135556 Squares of numbers not divisible by 3: a(n) = A001651(n)^2.

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