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A106853 Expansion of 1/(1-x*(1-4*x)). 13
1, 1, -3, -7, 5, 33, 13, -119, -171, 305, 989, -231, -4187, -3263, 13485, 26537, -27403, -133551, -23939, 510265, 606021, -1435039, -3859123, 1881033, 17317525, 9793393, -59476707, -98650279, 139256549, 533857665, -23168531, -2158599191, -2065925067 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums of Riordan array (1,x(1-4x)). 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)).

For n>=1 a(n) equals the determinant of the n X n matrix with 2's along the superdiagonal and the subdiagonal, and 1's along the main diagonal, and 0's everywhere else. [John M. Campbell, Jun 04 2011]

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,-4).

FORMULA

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

a(n) = 2^n*(cos(2*n*atan(sqrt(15)/5))+sqrt(15)sin(2*n*atan(sqrt(15)/5))/15).

a(n) = ((1+sqrt(-15))^(n+1)-(1-sqrt(-15))^(n+1))/(2^(n+1)*sqrt(-15)).

a(n) = sum(k=0..n, (-1)^(n-k)*binomial(k, n-k)*4^(n-k) ).

a(n) = a(n-1)-4*a(n-2), a(0)=1, a(1)=1. [Philippe Deléham, Oct 21 2008]

a(n) = sum(k=0..n, A109466(n,k)*4^(n-k)). [Philippe Deléham, Oct 25 2008]

G.f.: 1/(1-2*x)^2/(1 + 3*x*G(0)/2), where G(k)= 1 + 1/(1 - x/(x + (k+1)/(2*k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013

MATHEMATICA

Join[{a=1, b=1}, Table[c=b-4*a; a=b; b=c, {n, 80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)

CoefficientList[Series[1/(1-x*(1-4x)), {x, 0, 40}], x] (* or *) LinearRecurrence[ {1, -4}, {1, 1}, 40] (* Harvey P. Dale, May 26 2013 *)

PROG

(Sage) [lucas_number1(n, 1, 4) for n in xrange(1, 36)] # [Zerinvary Lajos, Apr 22 2009]

CROSSREFS

Cf. A106852.

Sequence in context: A161818 A161509 A108974 * A083778 A107785 A277207

Adjacent sequences:  A106850 A106851 A106852 * A106854 A106855 A106856

KEYWORD

easy,sign

AUTHOR

Paul Barry, May 08 2005

STATUS

approved

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Last modified March 28 11:45 EDT 2017. Contains 284186 sequences.