A106852 Expansion of 1/(1-x*(1-3*x)).

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

Offset: 0

Author

Paul Barry, May 08 2005

Keywords

• sign
• easy

Comments

Row sums of Riordan array (1, x*(1-3*x)). 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-r*x)).

Row sums of Riordan array (1/(1+3*x^2), x/(1+3*x^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 = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-3).

Crossrefs

Cf. A214733.

Programs

Magma

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

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 *)

PARI

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

PARI

x='x+O('x^30); Vec(1/(1-x+3*x^2)) \\ G. C. Greubel, Jan 14 2018

Sage

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

Formula

From Paul Barry, Sep 10 2005: (Start)

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

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

a(n) = 3^(n/2)(cos(-n*arccot(sqrt(11)/11))-sqrt(11)*sin(-n*arccot(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_{k=0..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

G.f.: Sum_{n >= 0} x^n * Product_{k = 1..n} (k - 3*x)/(1 + k*x). - Peter Bala, Jul 06 2025