A110311 -id:A110311

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A110307

Expansion of (1+2*x)/((1+x+x^2)*(1+5*x+x^2)).

+10
5

1, -4, 17, -80, 384, -1842, 8827, -42292, 202631, -970862, 4651680, -22287540, 106786021, -511642564, 2451426797, -11745491420, 56276030304, -269634660102, 1291897270207, -6189851690932, 29657361184451, -142096954231322, 680827409972160, -3262040095629480

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

FORMULA

a(n+2) = - 5*a(n+1) - a(n) - A099837(n+1).

a(n) + a(n+1) + a(n+2) = A002320(n).

a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019

a(n) = (1/4)*(3*U(n,-5/2) + U(n-1,-5/2) + U(n,-1/2) - U(n-1,-1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 03 2023

MAPLE

seriestolist(series((1+2*x)/((x^2+x+1)*(x^2+5*x+1)), x=0, 25));

MATHEMATICA

LinearRecurrence[{-6, -7, -6, -1}, {1, -4, 17, -80}, 41] (* G. C. Greubel, Jan 03 2023 *)

PROG

(PARI) Vec((1+2*x)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+2*x)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

(SageMath)

def U(n, x): return chebyshev_U(n, x)

def A110307(n): return (1/4)*(3*U(n, -5/2) +U(n-1, -5/2) +U(n, -1/2) -U(n-1, -1/2))

[A110307(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

CROSSREFS

Cf. A002320, A004253, A049347, A099837, A110308, A110309, A110310, A110311.

KEYWORD

easy,sign

AUTHOR

Creighton Dement, Jul 19 2005

STATUS

approved

A110308

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

+10
5

0, -2, 11, -52, 247, -1182, 5664, -27140, 130037, -623044, 2985181, -14302860, 68529120, -328342742, 1573184591, -7537580212, 36114716467, -173036002122, 829065294144, -3972290468600, 19032387048857, -91189644775684, 436915836829561, -2093389539372120

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

FORMULA

a(n+2) = - 5*a(n+1) - a(n) - A099837(n+2).

a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019

a(n) = (1/4)*(2*U(n, -5/2) + U(n-1, -5/2) - 2*U(n, -1/2) - U(n-1, -1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 03 2023

MAPLE

seriestolist(series(-x*(2+x)/((x^2+x+1)*(x^2+5*x+1)), x=0, 25));

MATHEMATICA

LinearRecurrence[{-6, -7, -6, -1}, {0, -2, 11, -52}, 40] (* G. C. Greubel, Jan 03 2023 *)

PROG

(PARI) concat(0, Vec(-x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25))) \\ Colin Barker, Apr 30 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( -x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

(SageMath)

def U(n, x): return chebyshev_U(n, x)

def A110308(n): return (1/4)*(2*U(n, -5/2) +U(n-1, -5/2) -2*U(n, -1/2) -U(n-1, -1/2))

[A110308(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

CROSSREFS

Cf. A004253, A049347, A099837, A110307, A110309, A110310, A110311.

KEYWORD

easy,sign

AUTHOR

Creighton Dement, Jul 19 2005

STATUS

approved

A110309

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

+10
5

1, -3, 12, -57, 275, -1320, 6325, -30303, 145188, -695637, 3332999, -15969360, 76513801, -366599643, 1756484412, -8415822417, 40322627675, -193197315960, 925663952125, -4435122444663, 21249948271188, -101814618911277, 487823146285199, -2337301112514720

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

FORMULA

a(n+2) = - 5*a(n+1) - a(n) + (-1)^n*A109265(n+3).

a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019

a(n) = (1/2)*(ChebyshevU(n, -5/2) + ChebyshevU(n, -1/2)). - G. C. Greubel, Jan 03 2023

MAPLE

seriestolist(series((1+3*x+x^2)/((x^2+5*x+1)*(x^2+x+1)), x=0, 25));

MATHEMATICA

LinearRecurrence[{-6, -7, -6, -1}, {1, -3, 12, -57}, 40] (* G. C. Greubel, Jan 03 2023 *)

PROG

(PARI) Vec((1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

(SageMath)

def A110309(n): return (1/2)*(chebyshev_U(n, -5/2)+chebyshev_U(n, -1/2))

[A110309(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

CROSSREFS

Cf. A004253, A049347, A109265, A110307, A110308, A110310, A110311.

KEYWORD

easy,sign

AUTHOR

Creighton Dement, Jul 19 2005

STATUS

approved

A110310

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

+10
5

1, -7, 36, -173, 827, -3960, 18973, -90907, 435564, -2086913, 9998999, -47908080, 229541401, -1099798927, 5269453236, -25247467253, 120967883027, -579591947880, 2776991856373, -13305367333987, 63749844813564, -305443856733833, 1463469438855599, -7011903337544160

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

FORMULA

a(n+2) = - 5*a(n+1) - a(n) - (-1)^n*A109265(n+3).

a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019

a(n) = (1/2)*(3*ChevyshevU(n, -5/2) - ChebyshevU(n, -1/2)). - G. C. Greubel, Jan 02 2023

MAPLE

seriestolist(series((1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)), x=0, 25));

MATHEMATICA

LinearRecurrence[{-6, -7, -6, -1}, {1, -7, 36, -173}, 40] (* G. C. Greubel, Jan 02 2023 *)

PROG

(PARI) Vec((1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 02 2023

(SageMath)

def U(n, x): return chebyshev_U(n, x)

def A110310(n): return (1/2)*(3*U(n, -5/2) - U(n, -1/2))

[A110310(n) for n in range(41)] # G. C. Greubel, Jan 02 2023

CROSSREFS

Cf. A004253, A049347, A109265, A110307, A110308, A110309, A110311.

KEYWORD

easy,sign

AUTHOR

Creighton Dement, Jul 19 2005

STATUS

approved

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