A112712 -id:A112712 - cp's OEIS frontend

A112713 Expansion of x/(1 - x + x^5 - x^6).

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

Offset: 0

Paul Barry, Sep 15 2005

Decimal expansion of 10000/900009. - Elmo R. Oliveira, May 08 2024

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,-1,1).

Cf. A000035, A133872, A088911, A131078, A112712.

CoefficientList[Series[x/(1 - x + x^5 - x^6), {x, 0, 100}], x] (* or *) LinearRecurrence[{1, 0, 0, 0, -1, 1}, {0, 1, 1, 1, 1, 1}, 100] (* Harvey P. Dale, Feb 16 2014 *)

G.f.: x/(1 - x + x^5 - x^6) = x/((1-x)*(1+x)*(1-x+x^2-x^3+x^4)).
a(n) = a(n-1) - a(n-5) + a(n-6).
a(n) = a(n-10).
a(n) = (Sum_{k=0..floor((n+2)/2)} (-1)^(k+1)*C(n-k+2, k-1)*F(n-2*k+2)) mod 2.
a(n) = A112712(n) mod 2.

Incorrect g.f. removed by Georg Fischer, May 15 2019

A129920 Expansion of -1/(1 - x + 3x^2 - 2x^3 + x^4 - 2*x^5 + x^6).

Original entry on oeis.org

-1, -1, 2, 3, -4, -10, 5, 29, 2, -76, -45, 178, 212, -361, -750, 565, 2282, -306, -6206, -2428, 15176, 14353, -32719, -55104, 57933, 176234, -61524, -499047, -97429, 1271400, 921652, -2887641, -3948938, 5590078, 13380187, -7828378, -39536779, 108416, 104810904

Offset: 0

Roger L. Bagula, Jun 05 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-3,2,-1,2,-1).

Cf. A008620, A010892, A014019, A099443, A099479, A099480, A112712.

Cf. A125629, A129704, A129903.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) )); // G. C. Greubel, Sep 28 2024

CoefficientList[Series[-1/(1-x +3*x^2 -2*x^3 +x^4 -2*x^5 +x^6), {x,0,50}], x]

def A129920_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) ).list()
A129920_list(50) # G. C. Greubel, Sep 28 2024

a(n) = a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6), n >= 6. - Franck Maminirina Ramaharo, Jan 08 2019

Edited by Franck Maminirina Ramaharo, Jan 08 2019

A125629 Expansion of -1/(1 - x + x^2 - x^3 + x^4 + x^6).

Original entry on oeis.org

1, -1, 0, 0, 0, 1, 2, 2, 1, 0, -1, -3, -5, -5, -3, 0, 4, 9, 13, 13, 8, -1, -13, -26, -35, -34, -20, 6, 40, 74, 95, 89, 48, -26, -120, -209, -258, -232, -111, 98, 355, 587, 699, 601, 245, -342, -1040, -1641, -1887, -1545, -504, 1137, 3023, 4568, 5073, 3936, 912

Offset: 0

Roger L. Bagula, Jun 07 2007

The Knot Atlas, L6a3
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,0,-1).

Cf. A008620, A010892, A014019, A099443, A099479, A099480, A112712, A129920, A129704, A129903.

CoefficientList[Series[-1/(1 - x + x^2 - x^3 + x^4 + x^6), {x, 0, 50}], x]

G.f.: 1/(x^(17/2)*f(x)), where f(x) = -1/x^(5/2) - 1/x^(9/2) + 1/x^(11/2) + -1/x^(13/2) + 1/x^(15/2) - 1/x^(17/2) is the Jones polynomial for the link with Dowker-Thistlethwaite notation L6a3.

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) - a(n-6), n >= 6. - Franck Maminirina Ramaharo, Jan 08 2019

Edited by Franck Maminirina Ramaharo, Jan 08 2019

cp's OEIS Frontend

A112713 Expansion of x/(1 - x + x^5 - x^6).

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A129920 Expansion of -1/(1 - x + 3x^2 - 2x^3 + x^4 - 2*x^5 + x^6).

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A125629 Expansion of -1/(1 - x + x^2 - x^3 + x^4 + x^6).

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cp's OEIS Frontend

A112713 Expansion of x/(1 - x + x^5 - x^6).

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A129920 Expansion of -1/(1 - x + 3*x^2 - 2*x^3 + x^4 - 2*x^5 + x^6).

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A125629 Expansion of -1/(1 - x + x^2 - x^3 + x^4 + x^6).

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A129920 Expansion of -1/(1 - x + 3x^2 - 2x^3 + x^4 - 2*x^5 + x^6).