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URL: https://oeis.org/A176737

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A176737
Expansion of 1 / (1-4*x^2-3*x^3). (4,3)-Padovan sequence.
2
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.
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 * A370566 A046162 A060509
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 26 2010
STATUS
approved