VOOZH about

URL: https://oeis.org/A176738

⇱ A176738 - OEIS

login
A176738
Expansion of 1 / ((1+x)*(1-x-4*x^2)). (5,4)-Padovan sequence.
1
1, 0, 5, 4, 25, 40, 141, 300, 865, 2064, 5525, 13780, 35881, 91000, 234525, 598524, 1536625, 3930720, 10077221, 25800100, 66108985, 169309384, 433745325, 1110982860, 2845964161, 7289895600, 18673752245, 47833334644, 122528343625, 313861682200, 803975056701
OFFSET
0,3
COMMENTS
See A000931 (Padovan), and the W. Lang link given there.
FORMULA
O.g.f.: 1/((1-x-4*x^2)*(1+x)) = ((3-4*x)/(1-x-4*x^2) -1/(1+x))/2.
a(n) = (3*b(n) - 4*b(n-1) - (-1)^n)/2, n>=0, with b(n):=A006131(n) ((1,4)-Fibonacci), b(-1):=0.
From Colin Barker, Dec 25 2017: (Start)
a(n) = ((-1)^(1+n) + (2^(-1-n)*((1+sqrt(17))^n*(-5+3*sqrt(17)) + (1-sqrt(17))^n*(5+3*sqrt(17)))) / sqrt(17)) / 2.
a(n) = 5*a(n-2) + 4*a(n-3) for n>2.
(End)
MATHEMATICA
LinearRecurrence[{0, 5, 4}, {1, 0, 5}, 40] (* Harvey P. Dale, May 27 2016 *)
f[n_] := Simplify[((-1)^(1 +n) + (2^(-1 -n)*((1 + Sqrt[17])^n*(-5 +3Sqrt[17]) + (1 -Sqrt[17])^n*(5 + 3Sqrt[17])))/Sqrt[17])/2]; Array[f, 31, 0] (* or *)
CoefficientList[Series[1/(1 -5x^2 -4x^3), {x, 0, 30}], x] (* or *)
RecurrenceTable[{a[n] == 5 a[n - 2] + 4 a[n - 3], a[0] == 1, a[1] == 0, a[2] == 5}, a, {n, 30}] (* Robert G. Wilson v, Dec 25 2017 *)
PROG
(PARI) Vec(1 / ((1 + x)*(1 - x - 4*x^2)) + O(x^40)) \\ Colin Barker, Dec 25 2017
CROSSREFS
Cf. A176737 ((4,3)-Padovan).
Sequence in context: A180137 A288207 A038246 * A301824 A248255 A277058
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 14 2010
STATUS
approved