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A106852 Expansion of 1/(1-x(1-3x)). 18
1, 1, -2, -5, 1, 16, 13, -35, -74, 31, 253, 160, -599, -1079, 718, 3955, 1801, -10064, -15467, 14725, 61126, 16951, -166427, -217280, 282001, 933841, 87838, -2713685, -2977199, 5163856, 14095453, -1396115, -43682474, -39494129, 91553293, 210035680, -64624199, -694731239, -500858642 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums of Riordan array (1,x(1-3x)) In general, 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)).

Row sums of Riordan array (1/(1+3x^2),x/(1+3x^2)). - Paul Barry, Sep 10 2005

See A214733 for a differently signed version of this sequence. - Peter Bala, Nov 21 2016

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,-3).

FORMULA

From Paul Barry, Sep 10 2005: (Start)

G.f.: 1/(1-x+3x^2).

a(n) = 2*sqrt(33)*3^(n/2)*cos((n+1)*atan(sqrt(11)/11)-pi*n/2)/11.

a(n) = 3^(n/2)(cos(-n*acot(sqrt(11)/11))-sqrt(11)*sin(-n*acot(sqrt(11)/11))/11).

a(n) = ((1+sqrt(-11))^(n+1)-(1-sqrt(-11))^(n+1))/(2^(n+1)sqrt(-11)).

a(n) = Sum_{k=0..n, (-1)^(n-k)*binomial(k, n-k)*3^(n-k)} = sum{0<=k<=n} A109466(n,k)*3^(n-k).

a(n) = Sum_{k=0..n, C((n+k)/2, k)*(-3)^((n-k)/2)*(1+(-1)^(n-k))/2}.

a(n) = Sum_{k=0..floor(n/2), C(n-k, k)(-3)^k}. (End)

a(n) = a(n-1)-3*a(n-2), a(0)=1, a(1)=1. - Philippe Deléham, Oct 21 2008

G.f.: Q(0)/x -1/x, where Q(k) = 1 - 3*x^2 + (k+2)*x - x*(k+1 - 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013

MATHEMATICA

CoefficientList[Series[1/(1 - x (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 07 2013 *)

LinearRecurrence[{1, -3}, {1, 1}, 40] (* Harvey P. Dale, Apr 02 2016 *)

PROG

(Sage) [lucas_number1(n, 1, +3) for n in range(1, 40)] /* Zerinvary Lajos, Apr 22 2009 */

(PARI) a(n)=([0, 1; -3, 1]^n*[1; 1])[1, 1] \\ Charles R Greathouse IV, Nov 21 2016

CROSSREFS

Cf. A214733.

Sequence in context: A197365 A121579 A214733 * A162975 A187244 A120294

Adjacent sequences:  A106849 A106850 A106851 * A106853 A106854 A106855

KEYWORD

sign,easy

AUTHOR

Paul Barry, May 08 2005

STATUS

approved

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Last modified March 28 11:27 EDT 2017. Contains 284186 sequences.