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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
a(n) = 3*a(n-1) + a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=4.
a(n+1) = Sum_{0<=k<=0..n} A216182(n,k). - Philippe Deléham, Mar 11 2013
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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
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(MAGMAMagma) [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
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a(n) = T(2n+1), where T(n) are the tribonacci numbers A000073.
G. C. Greubel, <a href="/A073717/b073717.txt">Table of n, a(n) for n = 0..1000</a>
a(n) = 3*a(n-1) +a(n-2) +a(n-3), a(0)=0, a(1)=1, a(2)=4.
a(n) = A113300(n-1) + A113300(n). - R. J. Mathar, Jul 04 2019
(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
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