A176737 Expansion of 1 / (1-4*x^2-3*x^3). (4,3)-Padovan sequence.

1, 0, 4, 3, 16, 24, 73, 144, 364, 795, 1888, 4272, 9937, 22752, 52564, 120819, 278512, 640968, 1476505, 3399408, 7828924, 18027147, 41513920, 95595360, 220137121, 506923200, 1167334564, 2688104163, 6190107856, 14254420344, 32824743913, 75588004944, 174062236684

Offset

0,3

Comments

See A000931 (Padovan), and the W. Lang link given there.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,4,3).

Formula

O.g.f.: 1/((1-x-3*x^2)*(1+x)) = (2-3*x)/(1-x-3*x^2) -1/(1+x).

a(n) = 2*b(n) - 3*b(n-1) - (-1)^n, n>=0, with b(n):=A006130(n) ((1,3)-Fibonacci), b(-1):=0.

From Wolfdieter Lang, Aug 26 2010: (Start)

a(n) = a(n-1) + 3*a(n-2) + (-1)^n, n>=2, a(0)=1, a(1)=0.

Due to the identity for the o.g.f. A(x): A(x)= x*(1 + 3*x)*A(x) + 1/(1+x).

(This recurrence was observed by Gary Detlefs in an Aug 24 2010 email to the author.)

(End)

a(n) = 4*a(n-2) + 3*a(n-3) for n>2. - Harvey P. Dale, Jan 21 2013

a(n) = (-1)^(n+1)*A140165(n+2)-(-1)^n. - R. J. Mathar, Apr 22 2013

a(n) = ((-1)^(1+n) + (2^(-n)*((-2+sqrt(13))*(1+sqrt(13))^n + (1-sqrt(13))^n*(2+sqrt(13)))) / sqrt(13)). - Colin Barker, Dec 25 2017

Mathematica

CoefficientList[Series[1/(1-4*x^2-3*x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[ {0, 4, 3}, {1, 0, 4}, 40] (* Harvey P. Dale, Jan 21 2013 *)

Prog

(PARI) Vec(1 / ((1 + x)*(1 - x - 3*x^2)) + O(x^40)) \\ Colin Barker, Dec 25 2017

Crossrefs

Cf. A053088 ((3,2)-Padovan).

Sequence in context: A288595 A288067 A038233 * A046162 A060509 A113203

Adjacent sequences: A176734 A176735 A176736 * A176738 A176739 A176740

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 26 2010

Status

approved