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a(n) = T(3n+1), where T(n) are tribonacci numbers A000073.
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%I #14 May 31 2024 14:39:18

%S 0,2,13,81,504,3136,19513,121415,755476,4700770,29249425,181997601,

%T 1132436852,7046319384,43844049029,272809183135,1697490356184,

%U 10562230626642,65720971788709,408933139743937,2544489349890656

%N a(n) = T(3n+1), where T(n) are tribonacci numbers A000073.

%C In general, the trisection of a third-order linear recurrence with signature (x,y,z) will result in a third-order recurrence with signature (x^3 + 3*x*y + 3*z, -3*x*y*z + y^3 - 3*z^2, z^3). - _Gary Detlefs_, May 29 2024

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-5,1).

%F a(n) = 7*a(n-1) - 5*a(n-2) + a(n-3), a(0)=0, a(1)=2, a(2)=13.

%F G.f.: (2^x - x^2)/(1 - 7*x + 5x^2 - x^3).

%t CoefficientList[Series[(2*x-x^2)/(1-7*x+5*x^2-x^3), {x, 0, 40}], x]

%t LinearRecurrence[{7,-5,1},{0,2,13},30] (* _Harvey P. Dale_, Jul 22 2021 *)

%Y Cf. A000073.

%K easy,nonn

%O 0,2

%A Mario Catalani (mario.catalani(AT)unito.it), Aug 24 2002

%E Definition corrected by _David Scambler_, Oct 18 2010