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A106854 Expansion of 1/(1-x(1-5x)). 9
1, 1, -4, -9, 11, 56, 1, -279, -284, 1111, 2531, -3024, -15679, -559, 77836, 80631, -308549, -711704, 831041, 4389561, 234356, -21713449, -22885229, 85682016, 200108161, -228301919, -1228842724, -87333129, 6056880491, 6493546136, -23790856319, -56258586999, 62695694596, 343988629591 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums of Riordan array (1,x(1-5x)). In general, a(n)=sum{k=0..n,(-1)^(n-k)*binomial(k,n-k)*r^(n-k)} yields the row sums of the Riordan array (1,x(1-kx)).

LINKS

Table of n, a(n) for n=0..33.

FORMULA

a(n)=((1+sqrt(-19))^(n+1)-(1-sqrt(-19))^(n+1))/(2^(n+1)sqrt(-19)); a(n)=sum{k=0..n, (-1)^(n-k)*binomial(k, n-k)*5^(n-k)}. a(n)=5^(n/2)(cos(-n*acot(sqrt(19)/19))-sqrt(19)sin(-n*acot(sqrt(19)/19))/19).

a(n)=a(n-1)-5*a(n-2), a(0)=1, a(1)=1. [From Philippe Deléham, Oct 21 2008]

a(n)=Sum_{k, 0<=k<=n}A109466(n,k)*5^(n-k). [From Philippe Deléham, Oct 25 2008]

G.f.: Q(0)/2, where Q(k) = 1 + 1/( 1 - x*(2*k+1 -5*x)/( x*(2*k+2 -5*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2013

MATHEMATICA

Join[{a=1, b=1}, Table[c=b-5*a; a=b; b=c, {n, 80}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 22 2011*)

CoefficientList[Series[1/(1-x(1-5x)), {x, 0, 40}], x] (* or *) LinearRecurrence[ {1, -5}, {1, 1}, 40] (* Harvey P. Dale, Jan 21 2012 *)

PROG

(Sage) [lucas_number1(n, 1, 5) for n in xrange(1, 35)] # Zerinvary Lajos, Jul 16 2008

CROSSREFS

Cf. A106852, A106853.

Sequence in context: A179055 A002641 A085724 * A099458 A069219 A010413

Adjacent sequences:  A106851 A106852 A106853 * A106855 A106856 A106857

KEYWORD

easy,sign

AUTHOR

Paul Barry, May 08 2005

STATUS

approved

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Last modified April 16 10:15 EDT 2014. Contains 240577 sequences.