A131736 -id:A131736 - cp's OEIS frontend

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0

Offset: 0

Paul Curtz, Jan 26 2008

Dirichlet series for the non-principal character modulo 6: L(s,chi) = Sum_{n>=1} a(n)/n^s. For example L(1,chi) = A093766, L(2,chi) = A214552, and L(3,chi) = Pi^3/(18*sqrt(3)). See Jolley eq. (314) and arXiv:1008.2547 L(m=6,r=2,s). - R. J. Mathar, Jul 31 2010

			G.f. = x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=6, Chi_2(n).
L. B. W. Jolley, Summation of Series, Dover (1961).

R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015.
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Cf. A093766, A120325, A131719, A131720, A131735, A131736, A214552.

&cat[[0, 1, 0, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A134667:=n->[0, 1, 0, 0, 0, -1][(n mod 6)+1]: seq(A134667(n), n=0..100);
# Wesley Ivan Hurt, Jun 20 2016

a[ n_] := JacobiSymbol[-12, n]; (* Michael Somos, Apr 24 2014 *)
a[ n_] := {1, 0, 0, 0, -1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Apr 24 2014 *)
PadRight[{},120,{0,1,0,0,0,-1}] (* Harvey P. Dale, Aug 01 2021 *)

{a(n) = [0, 1, 0, 0, 0, -1][n%6+1]}; /* Michael Somos, Feb 10 2008 */

{a(n) = kronecker(-12, n)}; /* Michael Somos, Feb 10 2008 */

{a(n) = if( n < 0, -a(-n), if( n<1, 0, direuler(p=2, n, 1 / (1 - kronecker(-12, p) * X))[n]))}; /* Michael Somos, Aug 11 2009 */

Euler transform of length 6 sequence [0, 0, 0, -1, 0, 1]. - Michael Somos, Feb 10 2008
G.f.: x * (1 - x^4) / (1 - x^6) = x*(1+x^2) / (1 + x^2 + x^4) = x*(1+x^2) / ( (1+x+x^2)*(x^2-x+1) ).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3)) where f(u, v, w) = w * (2 + v - u^2 - 2*v^2) - 2 * u * v. - Michael Somos, Aug 11 2009
a(n) is multiplicative with a(p^e) = 0^e if p = 2 or p = 3, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos, Aug 11 2009
a(-n) = -a(n). a(n+6) = a(n). a(2*n) = a(3*n) = 0.
sqrt(3)*a(n) = sin(Pi*n/3) + sin(2*Pi*n/3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-2) + a(n-4) = 0 for n>3. - Wesley Ivan Hurt, Jun 20 2016
E.g.f.: 2*sin(sqrt(3)*x/2)*cosh(x/2)/sqrt(3). - Ilya Gutkovskiy, Jun 21 2016

A100286 Expansion of (1+2x^2-2x^3+2*x^4)/(1-x+x^2-x^3+x^4-x^5).

Original entry on oeis.org

1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2

Offset: 0

Paul Barry, Nov 11 2004

Period 6: repeat [1,1,2,0,0,2]. - G. C. Greubel, Feb 06 2023

Decimal expansion of 3394/30303. - Elmo R. Oliveira, May 11 2024

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1).

Cf. A010892, A049347, A100284, A131736.

[2 +(n mod 2)*(1-((n+2) mod 3)) -((n+1) mod 3): n in [0..100]]; // G. C. Greubel, Feb 06 2023

CoefficientList[Series[(1+2x^2-2x^3+2x^4)/(1-x+x^2-x^3+x^4-x^5),{x,0,100}],x] (* Harvey P. Dale, Mar 03 2019 *)
PadRight[{}, 120, {1,1,2,0,0,2}] (* G. C. Greubel, Feb 06 2023 *)

def A100286(n): return 2 +(n%2)*(1-((n-1)%3)) -((n+1)%3)
[A100286(n) for n in range(101)] # G. C. Greubel, Feb 06 2023

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5).
a(n) = (1/6)*(6 + 3*cos(Pi*n/3) - 3*cos(2*Pi*n/3) + sqrt(3)*sin(Pi*n/3) - 3*sqrt(3)*sin(2*Pi*n/3)).
a(n) = mod(A100284(n), 3).
From G. C. Greubel, Feb 06 2023: (Start)
a(n) = a(n-6).
a(n) = (1/2)*(2 + A010892(n) - A049347(n) - 2*A049347(n-1)).
a(n) = 2 + (n mod 2)*(1 - (n-1 mod 3)) - (n+1 mod 3). (End)
a(n) = 1 + A131736(n). - Elmo R. Oliveira, Jun 20 2024

cp's OEIS Frontend

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

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A100286 Expansion of (1+2x^2-2x^3+2*x^4)/(1-x+x^2-x^3+x^4-x^5).

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cp's OEIS Frontend

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

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Maple

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PARI

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A100286 Expansion of (1+2*x^2-2*x^3+2*x^4)/(1-x+x^2-x^3+x^4-x^5).

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A100286 Expansion of (1+2x^2-2x^3+2*x^4)/(1-x+x^2-x^3+x^4-x^5).