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A292423 a(n) = 82*a(n-1) + a(n-2), where a(0) = 0, a(1) = 1.

%I A292423 #32 Dec 20 2024 17:48:31
%S A292423 0,1,82,6725,551532,45232349,3709604150,304232772649,24950796961368,
%T A292423 2046269583604825,167819056652557018,13763208915093280301,
%U A292423 1128750950094301541700,92571341116647819699701,7591978722515215516917182,622634826587364320206908625
%N A292423 a(n) = 82*a(n-1) + a(n-2), where a(0) = 0, a(1) = 1.
%C A292423 Every fifth term of A000129 is divisible by 29. Dividing every fifth term by 29 gives this sequence.
%H A292423 Colin Barker, <a href="/A292423/b292423.txt">Table of n, a(n) for n = 0..500</a>
%H A292423 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (82,1).
%F A292423 a(n) = A000129(5*n)/29.
%F A292423 From _Colin Barker_, Sep 20 2017: (Start)
%F A292423 G.f.: x / (1 - 82*x - x^2).
%F A292423 a(n) = (((-41-29*sqrt(2))^(-n)*(-1 + (-3363-2378*sqrt(2))^n))) / (58*sqrt(2)).
%F A292423 (End)
%p A292423 a:= n-> (<<0|1>, <1|82>>^n)[1, 2]:
%p A292423 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 18 2017
%t A292423 CoefficientList[Series[x/(1-82*x-x^2), {x,0,20}], x] (* _G. C. Greubel_, Feb 02 2019 *)
%t A292423 LinearRecurrence[{82,1},{0,1},20] (* _Harvey P. Dale_, Dec 20 2024 *)
%o A292423 (PARI) a(n) = ([82, 1; 1, 0]^n)[2, 1]; \\ _Altug Alkan_, Sep 18 2017
%o A292423 (PARI) concat(0, Vec(x / (1 - 82*x - x^2) + O(x^20))) \\ _Colin Barker_, Sep 20 2017
%o A292423 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x/(1-82*x-x^2) )); // _G. C. Greubel_, Feb 02 2019
%o A292423 (Sage) (x/(1-82*x-x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 02 2019
%Y A292423 Cf. A000129.
%K A292423 nonn,easy
%O A292423 0,3
%A A292423 _Bobby Jacobs_, Sep 18 2017