A099841 Expansion of (1-16*x)/(1-20*x+80*x^2).

1, 4, 0, -320, -6400, -102400, -1536000, -22528000, -327680000, -4751360000, -68812800000, -996147200000, -14417920000000, -208666624000000, -3019898880000000, -43704647680000000, -632501043200000000, -9153649049600000000, -132472897536000000000

Offset

0,2

Comments

Tenth binomial transform of A099839. Fifteenth binomial transform of A099840.

Links

G. C. Greubel, Table of n, a(n) for n = 0..850

Index entries for linear recurrences with constant coefficients, signature (20,-80).

Formula

G.f.: (1-16*x)/(1-20*x+80*x^2).

a(n) = (10 - 2*sqrt(5))^n*(1/2 + 3*sqrt(5)/10) + (10 + 2*sqrt(5))^n*(1/2 - 3*sqrt(5)/10).

a(n) = 20*a(n-1) - 80*a(n-2); a(0)=1, a(1)=4. - Harvey P. Dale, Jun 04 2013

a(n) = 2^(2*n-1)*5^((n-1)/2)*(-sqrt(5)*(1+(-1)^n)*Fibonacci(n-2) - (1 - (-1)^n)*Lucas(n-2)). - G. C. Greubel, Apr 21 2023

Mathematica

CoefficientList[Series[(1-16x)/(1-20x+80x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{20, -80}, {1, 4}, 30] (* Harvey P. Dale, Jun 04 2013 *)

Prog

(Magma) [n le 2 select 3*n-2 else 20*(Self(n-1) - 4*Self(n-2)): n in [1..41]]; // G. C. Greubel, Apr 21 2023
(SageMath)
def A099841(n): return 4^n*5^((n-1)/2)*(-sqrt(5)*((n-1)%2)*fibonacci(n-2) - (n%2)*lucas_number2(n-2, 1, -1))
[A099841(n) for n in range(41)] # G. C. Greubel, Apr 21 2023

Crossrefs

Cf. A000032, A000045, A099839, A099840.

Sequence in context: A119010 A255018 A176050 * A013452 A013453 A012688

Adjacent sequences: A099838 A099839 A099840 * A099842 A099843 A099844

Keyword

easy,sign

Author

Paul Barry, Oct 27 2004

Status

approved