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
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0,4,3).
Crossrefs
Cf. A053088 ((3,2)-Padovan).
Programs
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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 *) -
PARI
Vec(1 / ((1 + x)*(1 - x - 3*x^2)) + O(x^40)) \\ Colin Barker, Dec 25 2017
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
Comments