A100304 -id:A100304 - cp's OEIS frontend

A100303 Expansion of (1 - x - 4x^2)/(1 - x - 8x^2).

1, 0, 4, 4, 36, 68, 356, 900, 3748, 10948, 40932, 128516, 455972, 1484100, 5131876, 17004676, 58059684, 194097092, 658574564, 2211351300, 7479947812, 25170758212, 85010340708, 286376406404, 966459132068, 3257470383300, 10989143439844, 37048906506244, 124962054024996

Offset: 0

Paul Barry, Nov 12 2004

Construct a graph as follows: form the graph whose adjacency matrix is the tensor product of that of P_3 and [1,1;1,1], then add a loop at each of the extremity nodes. (Spectrum : [0; 1^3; (1-sqrt(33))/2;(1+sqrt(33))/2]). a(n) counts closed walks of length n at each of the internal nodes.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,8).

Cf. A015443, A100302, A100304.

[1] cat [n le 2 select 4*(n-1) else Self(n-1) +8*Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 04 2023

CoefficientList[Series[(1-x-4x^2)/(1-x-8x^2),{x,0,30}],x] (* Harvey P. Dale, Dec 01 2013 *)
LinearRecurrence[{1,8}, {1,0,4}, 31] (* G. C. Greubel, Feb 04 2023 *)

def A100303(n): return (1/2)*int(n==0) + 4*lucas_number1(n-1, 1, -8)
[A100303(n) for n in range(31)] # G. C. Greubel, Feb 04 2023

a(n) = 0^n/2 + (4/sqrt(33))*( ((1 + sqrt(33))/2)^(n-1) - ((1 - sqrt(33))/2)^(n-1) ).
a(n) = (1/2)*(A015443(n) - A015443(n-1)), n > 0. - Ralf Stephan, Jul 21 2013
E.g.f.: (33 + exp(x/2)*(33*cosh(sqrt(33)*x/2) - sqrt(33)*sinh(sqrt(33)*x/2)))/66. - Stefano Spezia, Sep 08 2022
a(n) = (1/2)*[n=0] + 4*A015443(n). - G. C. Greubel, Feb 04 2023

A100302 Expansion of (1 - x - 6x^2)/((1 - x)(1 - x - 8*x^2)).

Original entry on oeis.org

1, 1, 3, 5, 23, 57, 235, 685, 2559, 8033, 28499, 92757, 320743, 1062793, 3628731, 12131069, 41160911, 138209457, 467496739, 1573172389, 5313146295, 17898525401, 60403695755, 203591898957, 686821464991, 2315556656641, 7810128376563, 26334581629685, 88815608642183

Offset: 0

Paul Barry, Nov 12 2004

Construct a graph as follows: form the graph whose adjacency matrix is the tensor product of that of P_3 and [1,1;1,1], then add a loop at each of the extremity nodes. a(n) counts closed walks of length n at each of the extremity nodes.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,-8).

Cf. A015443, A100303, A100304.

I:=[1,1,3]; [n le 3 select I[n] else 2*Self(n-1) +7*Self(n-2) -8*Self(n-3): n in [1..31]]; // G. C. Greubel, Feb 04 2023

LinearRecurrence[{2,7,-8},{1,1,3},29] (* Stefano Spezia, Sep 08 2022 *)

def A100302(n): return (3 + lucas_number1(n+1, 1, -8))/4
[A100302(n) for n in range(31)] # G. C. Greubel, Feb 04 2023

a(n) = 2*a(n-1) + 7*a(n-2) - 8*a(n-3).
a(n) = 3/4 + (1/(4*sqrt(33)))*(((1 + sqrt(33))/2)^(n+1) - ((1 - sqrt(33))/2)^(n+1)).
E.g.f.: 3*exp(x)/4 + exp(x/2)*(33*cosh(sqrt(33)*x/2) + sqrt(33)*sinh(sqrt(33)*x/2))/132. - Stefano Spezia, Sep 08 2022
a(n) = (1/4)*(3 + A015443(n+1)). - G. C. Greubel, Feb 04 2023

A100305 Expansion of (1 - x - 4x^2)/(1 - 2x - 7x^2 + 8x^3).

Original entry on oeis.org

1, 1, 5, 9, 45, 113, 469, 1369, 5117, 16065, 56997, 185513, 641485, 2125585, 7257461, 24262137, 82321821, 276418913, 934993477, 3146344777, 10626292589, 35797050801, 120807391509, 407183797913, 1373642929981, 4631113313281, 15620256753125, 52669163259369, 177631217284365

Offset: 0

Paul Barry, Nov 12 2004

Construct a graph as follows: form the graph whose adjacency matrix is the tensor product of that of P_3 and [1,1;1,1], then add a loop at each of the 'internal' nodes. (Spectrum : [0^3; 1; (1-sqrt(33))/2;(1+sqrt(33))/2]). a(n) counts closed walks of length n at each of the 'internal' nodes.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,-8).

Cf. A015443, A100304.

Partial sums of A100303.

I:=[1,1,5]; [n le 3 select I[n] else 2*Self(n-1) +7*Self(n-2) -8*Self(n-3): n in [1..41]]; // G. C. Greubel, Feb 03 2023

CoefficientList[Series[(1-x-4x^2)/(1-2x-7x^2+8x^3),{x,0,40}],x] (* or *) LinearRecurrence[{2,7,-8},{1,1,5},40] (* Harvey P. Dale, Oct 05 2012 *)

def A100305(n): return (1/2)*(1 + lucas_number1(n+1, 1, -8))
[A100305(n) for n in range(41)] # G. C. Greubel, Feb 03 2023

a(n) = 2*a(n-1) + 7*a(n-2) - 8*a(n-3).
a(n) = 1/2 + 2^(-n)*(sqrt(33)/132)*((1 + sqrt(33))^(n+1) - (1 - sqrt(33))^(n+1)).
E.g.f.: exp(x)/2 + exp(x/2)*(33*cosh(sqrt(33)*x/2) + sqrt(33)*sinh(sqrt(33)*x/2))/66. - Stefano Spezia, Sep 08 2022
a(n) = (1/2)*(1 + (2*sqrt(2)*i)^n*ChebyshevU(n, -i/(4*sqrt(2)))). - G. C. Greubel, Feb 03 2023

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A100303 Expansion of (1 - x - 4x^2)/(1 - x - 8x^2).

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A100302 Expansion of (1 - x - 6x^2)/((1 - x)(1 - x - 8*x^2)).

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A100305 Expansion of (1 - x - 4x^2)/(1 - 2x - 7x^2 + 8x^3).

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cp's OEIS Frontend

A100303 Expansion of (1 - x - 4*x^2)/(1 - x - 8*x^2).

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A100302 Expansion of (1 - x - 6*x^2)/((1 - x)*(1 - x - 8*x^2)).

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A100305 Expansion of (1 - x - 4*x^2)/(1 - 2*x - 7*x^2 + 8*x^3).

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A100303 Expansion of (1 - x - 4x^2)/(1 - x - 8x^2).

A100302 Expansion of (1 - x - 6x^2)/((1 - x)(1 - x - 8*x^2)).

A100305 Expansion of (1 - x - 4x^2)/(1 - 2x - 7x^2 + 8x^3).