A168637 - OEIS

A168637

a(n) = a(n-1) + a(n-2) - a(n-4) starting a(0)=0, a(1)=1, a(2)=a(3)=3.

0, 1, 3, 3, 6, 8, 11, 16, 21, 29, 39, 52, 70, 93, 124, 165, 219, 291, 386, 512, 679, 900, 1193, 1581, 2095, 2776, 3678, 4873, 6456, 8553, 11331, 15011, 19886, 26344, 34899, 46232, 61245, 81133, 107479, 142380, 188614, 249861, 330996, 438477, 580859, 769475

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

REFERENCES

R. Pallu de la Barriere, Optimal Control Theory, Dover Publications, New York, 1967, pages 339-344

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

FORMULA

Limit_{n -> oo} a(n+1)/a(n) = A060006 (also a limiting value of A000931).

G.f.: x*(1 + 2*x - x^2)/((1-x)*(1 - x^2 - x^3)). [Dec 03 2009]

a(n) = 3*A000931(n+4) + 2*A000931(n+3) - 2. [Dec 03 2009]

a(n) = a(n-2) + a(n-3) + 2. - Greg Dresden, May 18 2020

MATHEMATICA

LinearRecurrence[{1, 1, 0, -1}, {0, 1, 3, 3}, 50] (* or *) CoefficientList[ Series[ x*(-1-2x+x^2)/((1-x)(x^3+x^2-1)), {x, 0, 50}], x] (* Harvey P. Dale, Jun 22 2011 *)

PROG

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 1, 1]^n*[0; 1; 3; 3])[1, 1] \\ Charles R Greathouse IV, Jul 29 2016

(Magma)

R<x>:=PowerSeriesRing(Integers(), 60);

[0] cat Coefficients(R!( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) )); // G. C. Greubel, Apr 20 2025

(SageMath)

def A168637_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) ).list()

print(A168637_list(60)) # G. C. Greubel, Apr 20 2025

CROSSREFS

Cf. A007307 (for a different starting vector of the Mma program).