A099479 -id:A099479 - cp's OEIS frontend

A099480 Count from 1, repeating 2*n five times.

1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 19, 20, 20, 20, 20, 20, 21, 22, 22, 22, 22, 22, 23, 24, 24, 24, 24, 24, 25, 26, 26, 26, 26, 26

Offset: 0

Paul Barry, Oct 18 2004

Could be called the Jones sequence of the knot 9_43, since the g.f. is the reciprocal of (a parameterization of) the Jones polynomial for 9_43.

Half the domination number of the knight's graph on a 2 X (n+1) chessboard. - David Nacin, May 28 2017

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A099479, A287393.

I:=[1,2,2,2,2,2]; [n le 6 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-2*Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..100]]; // Vincenzo Librandi, Sep 09 2015

LinearRecurrence[{2, -2, 2, -2, 2, -1}, {1, 2, 2, 2, 2, 2}, 100] (* Vincenzo Librandi, Sep 09 2015 *)
Table[If[EvenQ[n],{n,n,n,n,n},n],{n,30}]//Flatten (* Harvey P. Dale, Dec 15 2020 *)

G.f.: 1/((1-x+x^2)(1-x-x^3+x^4)) = 1/(1-2x+2x^2-2x^3+2x^4-2x^5+x^6);
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+2*a(n-5)-a(n-6), n>5;
a(n) = -cos(Pi*2n/3+Pi/3)/6+sqrt(3)*sin(Pi*2n/3+Pi/3)/18-sqrt(3)*cos(Pi*n/3+Pi/6)/6+sin(Pi*n/3+Pi/6)/2+(n+3)/3.
a(n) = Sum_{i=0..n+1} floor((i-1)/6) - floor((i-3)/6). - Wesley Ivan Hurt, Sep 08 2015
a(n) = A287393(n+1)/2. - David Nacin, May 28 2017

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

A099480 Count from 1, repeating 2*n five times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

cp's OEIS Frontend

A099480 Count from 1, repeating 2*n five times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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