A099504 -id:A099504 - cp's OEIS frontend

A099503 Expansion of 1/(1-4*x+x^3).

1, 4, 16, 63, 248, 976, 3841, 15116, 59488, 234111, 921328, 3625824, 14269185, 56155412, 220995824, 869714111, 3422701032, 13469808304, 53009519105, 208615375388, 820991693248, 3230957253887, 12715213640160, 50039862867392

Offset: 0

Paul Barry, Oct 20 2004

A transform of A000302 under the mapping g(x) ->(1/(1+x^3)) * g(x/(1+x^3)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,-1).

Cf. A000071, A000302, A076264, A099504.

[n le 3 select 4^(n-1) else 4*Self(n-1) -Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 03 2023

CoefficientList[Series[1/(1-4x+x^3),{x,0,30}],x]  (* Harvey P. Dale, Apr 01 2011 *)
LinearRecurrence[{4,0,-1}, {1,4,16}, 30] (* G. C. Greubel, Aug 03 2023 *)

@CachedFunction
def a(n): # a = A099503
    if (n<3): return 4^n
    else: return 4*a(n-1) - a(n-3)
[a(n) for n in range(31)] # G. C. Greubel, Aug 03 2023

a(n) = 4*a(n-1) - a(n-3).

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k*4^(n-3*k).

A099505 A transform of the Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 1, -3, -2, -8, -5, -12, -4, -11, 6, -1, 28, 16, 56, 26, 72, 3, 51, -80, -19, -224, -104, -373, -99, -408, 154, -205, 770, 222, 1655, 540, 2367, 24, 2196, -2196, 664, -6440, -1555, -11543, -1750, -14427, 4690, -11340, 22009, -1595, 49534, 7008, 77051, -5264, 85296, -65467, 57874

Offset: 0

Paul Barry, Oct 20 2004

A transform of A000045 under the mapping g(x) -> (1/(1+x^3)) * g(x/(1+x^3)).

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

Cf. A000045, A099503, A099504.

I:=[0,1,1,2,1,2]; [n le 6 select I[n] else Self(n-1) +Self(n-2) -2*Self(n-3) +Self(n-4) -Self(n-6): n in [1..65]]; // G. C. Greubel, Aug 03 2023

LinearRecurrence[{1,1,-2,1,0,-1}, {0,1,1,2,1,2}, 65] (* G. C. Greubel, Aug 03 2023 *)

@CachedFunction
def a(n): # a = A099505
    if (n<6): return (0,1,1,2,1,2)[n]
    else: return a(n-1) +a(n-2) -2*a(n-3) +a(n-4) -a(n-6)
[a(n) for n in range(71)] # G. C. Greubel, Aug 03 2023

G.f.: x/(1 - x - x^2 + 2*x^3 - x^4 + x^6).
a(n) = a(n-1) + a(n-2) - 2*a(n-3) + a(n-4) - a(n-6).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k*Fibonacci(n-3*k).

A144701 Hankel transform of expansion of 1/c(x)^3, c(x) the g.f. of A000108.

Original entry on oeis.org

1, -9, 26, -25, -36, 133, -132, -81, 375, -374, -144, 806, -805, -225, 1480, -1479, -324, 2451, -2450, -441, 3773, -3772, -576, 5500, -5499, -729, 7686, -7685, -900, 10385, -10384, -1089, 13651, -13650, -1296, 17538, -17537

Offset: 0

Paul Barry, Sep 19 2008

Hankel transform of A115142.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,-10,-16,-19,-16,-10,-4,-1).

Cf. A000108, A099504, A115142, A128504.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)*(1-5*x+x^2)/(1+x+x^2)^4 )); // G. C. Greubel, Jun 16 2022

LinearRecurrence[{-4,-10,-16,-19,-16,-10,-4,-1}, {1,-9,26,-25,-36,133,-132,-81}, 40] (* G. C. Greubel, Jun 16 2022 *)

def A144701_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)*(1-5*x+x^2)/(1+x+x^2)^4 ).list()
A144701_list(40) # G. C. Greubel, Jun 16 2022

G.f.: (1+x)*(1-x)*(1-5*x+x^2)/(1+x+x^2)^4.

a(n) = (6 - 7*n - 9*n^2 - 2*n^3)*cos(2*Pi*n/3)/6 - sqrt(3)*(42 + 55*n + 21*n^2 + 2*n^3)*sin(2*Pi*n/3)/18.

A099524 Expansion of 1/(1-5*x-x^3).

Original entry on oeis.org

1, 5, 25, 126, 635, 3200, 16126, 81265, 409525, 2063751, 10400020, 52409625, 264111876, 1330959400, 6707206625, 33800145001, 170331684405, 858365628650, 4325628288251, 21798473125660, 109850731256950, 553579284573001

Offset: 0

Paul Barry, Oct 20 2004

A transform of A000351 under the mapping mapping g(x)->(1/(1-x^3))g(x/(1-x^3)).

a(n) equals the number of n-length words on {0,1,2,3,4,5} such that 0 appears only in a run which length is a multiple of 3. - Milan Janjic, Feb 17 2015

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
Index entries for linear recurrences with constant coefficients, signature (5,0,1).

Cf. A000351, A099504.

CoefficientList[Series[1/(1-5x-x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {5,0,1},{1,5,25},30] (* Harvey P. Dale, May 08 2012 *)

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

a(n) = Sum_(k=0..floor(n/3)) binomial(n-2*k, k)*5^(n-3*k).

cp's OEIS Frontend

A099503 Expansion of 1/(1-4*x+x^3).

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A099505 A transform of the Fibonacci numbers.

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A144701 Hankel transform of expansion of 1/c(x)^3, c(x) the g.f. of A000108.

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A099524 Expansion of 1/(1-5*x-x^3).

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