A099480 -id:A099480 - cp's OEIS frontend

A099479 Count, repeating 4n three times for n > 0.

0, 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 17, 18, 19, 20, 20, 20, 21, 22, 23, 24, 24, 24, 25, 26, 27, 28, 28, 28, 29, 30, 31, 32, 32, 32, 33, 34, 35, 36, 36, 36, 37, 38, 39, 40, 40, 40, 41, 42, 43, 44, 44, 44, 45, 46, 47, 48, 48, 48, 49

Offset: 0

Paul Barry, Oct 18 2004

A Chebyshev transform of A000975.
The denominator in the g.f. is 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + x^6, a version of the Jones polynomial of the knot 9_43.
The g.f. is the image of x/((1-x)*(1-x-2x^2)) under the Chebyshev transform A(x)->(1/(1+x^2))*A(x/(1+x^2)).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A099480.

I:=[0,1,2,3,4,4]; [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 25 2013

CoefficientList[Series[x (1 + x^2)/((1 - x + x^2) (1 - x - x^3 + x^4)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 25 2013 *)
LinearRecurrence[{2,-2,2,-2,2,-1},{0,1,2,3,4,4},80] (* Harvey P. Dale, Dec 11 2014 *)

x='x+O('x^50); concat([0], vec(x*(1+x^2)/((1-x+x^2)*(1-x-x^3+x^4)))) \\ G. C. Greubel, Oct 10 2017

G.f.: x*(1+x^2)/((1-x+x^2)*(1-x-x^3+x^4)).
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).
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 + 2(n+1)/3.
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(4*2^(n-1-2k)/3 - (-1)^n/6 - 1/2).
a(n) = floor((n+2)/3) + floor((n+3)/6) + floor((n+4)/6). - Ridouane Oudra, Jan 22 2024

A287393 Domination number for knight graph on a 2 X n board.

Original entry on oeis.org

0, 2, 4, 4, 4, 4, 4, 6, 8, 8, 8, 8, 8, 10, 12, 12, 12, 12, 12, 14, 16, 16, 16, 16, 16, 18, 20, 20, 20, 20, 20, 22, 24, 24, 24, 24, 24, 26, 28, 28, 28, 28, 28, 30, 32, 32, 32, 32, 32, 34, 36, 36, 36, 36, 36, 38, 40, 40, 40, 40, 40, 42, 44, 44, 44, 44, 44, 46

Offset: 0

David Nacin, May 24 2017

Minimum number of knights required to dominate a 2 X n board.

			For n=3 we need a(3)=4 knights to dominate a 2 X 3 board.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Knight (chess)
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A099480, A287394.

Table[2*(Floor[(i+4)/6]+Floor[(i+5)/6]), {i, 0, 67}]
LinearRecurrence[{2,-2,2,-2,2,-1},{0,2,4,4,4,4},70] (* Harvey P. Dale, Jul 07 2020 *)

concat(0, Vec(2*x / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, May 27 2017

[2*((i+4)//6+(i+5)//6) for i in range(68)]

a(n) = 2*(floor((n+4)/6) + floor((n+5)/6)).
From Colin Barker, May 26 2017: (Start)
G.f.: 2*x / ((1 - x)^2*(1 - x + x^2)*(1 + x + x^2)).
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) for n>5.
(End)
a(n) = 2*A099480(n-1).

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

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A099479 Count, repeating 4n three times for n > 0.

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A287393 Domination number for knight graph on a 2 X n board.

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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).