OFFSET
0,3
COMMENTS
In general, the bisection of a third-order linear recurrence with signature (x,y,z) will result in a third-order recurrence with signature (x^2 + 2*y, 2*z*x - y^2, z^2). - Gary Detlefs, May 29 2024
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,1).
FORMULA
a(n) = 3*a(n-1) + a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=4.
G.f.: x*(1+x)/(1-3*x-x^2-x^3).
a(n+1) = Sum_{k=0..n} A216182(n,k). - Philippe Deléham, Mar 11 2013
MATHEMATICA
CoefficientList[Series[(x+x^2)/(1-3x-x^2-x^3), {x, 0, 30}], x]
LinearRecurrence[{3, 1, 1}, {0, 1, 4}, 30] (* Harvey P. Dale, Sep 07 2015 *)
PROG
(Magma) [n le 3 select (n-1)^2 else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Nov 19 2021
(Sage)
def A073717_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(1+x)/(1-3*x-x^2-x^3) ).list()
A073717_list(30) # G. C. Greubel, Nov 19 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Aug 05 2002
STATUS
approved