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Pell trisection: Pell(3*n+1), n >= 0.
2

%I #26 Oct 18 2024 23:21:46

%S 1,12,169,2378,33461,470832,6625109,93222358,1311738121,18457556052,

%T 259717522849,3654502875938,51422757785981,723573111879672,

%U 10181446324101389,143263821649299118,2015874949414289041,28365513113449345692,399133058537705128729,5616228332641321147898

%N Pell trisection: Pell(3*n+1), n >= 0.

%C For the general computation of the o.g.f.s for the trisection of a sequence, given by its real o.g.f., see a _Wolfdieter Lang_ comment under A187357.

%H Colin Barker, <a href="/A187361/b187361.txt">Table of n, a(n) for n = 0..850</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,1).

%F a(n) = Pell(3*n+1), n >= 0, with Pell(n):=A000129(n).

%F O.g.f.: (1-2*x)/(1-14*x-x^2).

%F a(n) = 14*a(n-1) + a(n-2), a(0)= 1, a(1)=12.

%F a(n) = (((7-5*sqrt(2))^n*(-1+sqrt(2))+(1+sqrt(2))*(7+5*sqrt(2))^n))/(2*sqrt(2)). - _Colin Barker_, Jan 25 2016

%t Table[Fibonacci[3n + 1, 2], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)

%t LinearRecurrence[{14,1},{1,12},20] (* _Harvey P. Dale_, Jul 06 2023 *)

%o (PARI) Vec((1-2*x)/(1-14*x-x^2) + O(x^20)) \\ _Colin Barker_, Jan 25 2016

%Y Cf. A142588 (Pell(3*n)), A187362 (Pell(3*n+2)).

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Mar 09 2011