A100303 -id:A100303 - cp's OEIS frontend

A015443 Generalized Fibonacci numbers: a(n) = a(n-1) + 8*a(n-2).

1, 1, 9, 17, 89, 225, 937, 2737, 10233, 32129, 113993, 371025, 1282969, 4251169, 14514921, 48524273, 164643641, 552837825, 1869986953, 6292689553, 21252585177, 71594101601, 241614783017, 814367595825, 2747285859961

Offset: 0

Olivier Gérard

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-1) counts walks of length n between adjacent nodes. - Paul Barry, Nov 12 2004
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 2, 9*a(n-2) equals the number of 9-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
Pisano period lengths: 1, 1, 6, 1, 24, 6, 16, 1, 6, 24, 110, 6, 56, 16, 24, 2, 16, 6, 60, 24, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook), p. 318
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (1,8)

Cf. A015442, A015441, A100302, A100303.

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+8*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

CoefficientList[Series[1/(1-x-8*x^2), {x,0,50}], x] (* G. C. Greubel, Apr 30 2017 *)

Vec(1/(1-x-8*x^2)+O(x^99)) \\ Charles R Greathouse IV, Feb 03 2014

[lucas_number1(n,1,-8) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = (((1+sqrt(33))/2)^(n+1) - ((1-sqrt(33))/2)^(n+1))/sqrt(33).
a(n) = Sum_{k=0..n} A109466(n,k)*(-8)^(n-k). - Philippe Deléham, Oct 26 2008
G.f.: 1/(1-x-8*x^2). - R. J. Mathar, Apr 07 2011
a(n) = (Sum_{1<=k<=n+1, k odd} C(n+1,k)*33^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014

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

A100304 Expansion of (1 - x - 6x^2)/(1 - x - 8x^2).

Original entry on oeis.org

1, 0, 2, 2, 18, 34, 178, 450, 1874, 5474, 20466, 64258, 227986, 742050, 2565938, 8502338, 29029842, 97048546, 329287282, 1105675650, 3739973906, 12585379106, 42505170354, 143188203202, 483229566034, 1628735191650, 5494571719922, 18524453253122, 62481027012498

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 extremity nodes.

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

Essentially half A100303.

Cf. A015443, A100302 (partial sums), A100305.

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

LinearRecurrence[{1,8},{1,0,2},27] (* Stefano Spezia, Sep 08 2022 *)

def A100304(n): return (3/4)*int(n==0) + 2*lucas_number1(n-1, 1, -8)
[A100304(n) for n in range(31)] # G. C. Greubel, Feb 04 2023

a(n) = 3*0^n/4 + (2/sqrt(33))*( ((1 + sqrt(33))/2)^(n-1) - ((1 - sqrt(33))/2)^(n-1) ).
E.g.f.: (99 + exp(x/2)*(33*cosh(sqrt(33)*x/2) - sqrt(33)*sinh(sqrt(33)*x/2)))/132. - Stefano Spezia, Sep 08 2022
a(n) = (3/4)*[n=0] + 2*A015443(n-2). - 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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A015443 Generalized Fibonacci numbers: a(n) = a(n-1) + 8*a(n-2).

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

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A100304 Expansion of (1 - x - 6*x^2)/(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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A100302 Expansion of (1 - x - 6x^2)/((1 - x)(1 - x - 8*x^2)).

A100304 Expansion of (1 - x - 6x^2)/(1 - x - 8x^2).

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