A176738 Expansion of 1 / ((1+x)*(1-x-4*x^2)). (5,4)-Padovan sequence.

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.

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,5,4).

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

Adjacent sequences: A176735 A176736 A176737 * A176739 A176740 A176741

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 14 2010

Status

approved