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A133407 a(n) = a(n-1) + 5*a(n-2) for n >= 2, a(0)=1, a(1)=2.

%I A133407 #52 Jan 20 2025 14:54:44
%S A133407 1,2,7,17,52,137,397,1082,3067,8477,23812,66197,185257,516242,1442527,
%T A133407 4023737,11236372,31355057,87536917,244312202,681996787,1903557797,
%U A133407 5313541732,14831330717,41399039377,115555692962,322550889847,900329354657,2513083803892
%N A133407 a(n) = a(n-1) + 5*a(n-2) for n >= 2, a(0)=1, a(1)=2.
%H A133407 Alois P. Heinz, <a href="/A133407/b133407.txt">Table of n, a(n) for n = 0..1000</a> (first 100 terms from Matt C. Anderson)
%H A133407 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,5).
%F A133407 G.f.: (1+x)/(1-x-5*x^2).
%F A133407 a(n) = Sum_{k=0..n+1} A122950(n+1,k)*4^(n+1-k). - _Philippe Deléham_, Jan 08 2008
%F A133407 a(n) = ((21 - 3*sqrt(21))/42)*(1/2 - (1/2)*sqrt(21))^n + ((21 + 3*sqrt(21))/42)*(1/2 + (1/2)*sqrt(21))^n. - _Richard Choulet_, Nov 20 2008
%p A133407 a:= n-> (<<0|1>, <5|1>>^n. <<1, 2>>)[1,1]:
%p A133407 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 20 2025
%t A133407 LinearRecurrence[{1,5},{1,2},30] (* _Harvey P. Dale_, Jul 23 2013 *)
%o A133407 (PARI) x='x+O('x^99); Vec((1+x)/(1-x-5*x^2)) \\ _Altug Alkan_, Aug 28 2017
%Y A133407 Cf. A030195 (shifted binomial transform).
%K A133407 nonn,easy
%O A133407 0,2
%A A133407 _Philippe Deléham_, Jan 03 2008