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A133294 a(n) = 2*a(n-1) + 10*a(n-2), a(0)=1, a(1)=1.

%I A133294 #35 Jan 01 2024 11:44:46
%S A133294 1,1,12,34,188,716,3312,13784,60688,259216,1125312,4842784,20938688,
%T A133294 90305216,389997312,1683046784,7266066688,31362601216,135385869312,
%U A133294 584397750784,2522654194688,10889285897216,47005113741312
%N A133294 a(n) = 2*a(n-1) + 10*a(n-2), a(0)=1, a(1)=1.
%C A133294 Binomial transform of [1, 0, 11, 0, 121, 0, 1331, 0, 14641, 0, ...]=: powers of 11 (A001020) with interpolated zeros. - _Philippe Deléham_, Dec 02 2008
%C A133294 A083101 is an essentially identical sequence (with a different start). - _N. J. A. Sloane_, Dec 31 2012
%H A133294 G. C. Greubel, <a href="/A133294/b133294.txt">Table of n, a(n) for n = 0..1000</a>
%H A133294 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,10).
%F A133294 a(n) = Sum_{k=0..n} A098158(n,k)*11^(n-k).
%F A133294 G.f.: (1-x)/(1-2*x-10*x^2).
%F A133294 a(n) = A083101(n-1) for n >= 1.
%F A133294 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 14 2013
%t A133294 a[n_]:= Simplify[((1+Sqrt[11])^n + (1-Sqrt[11])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-10x^2), {x,0,30}], x] (* Or *) LinearRecurrence[{2, 10}, {1, 1}, 30] (* _Robert G. Wilson v_, Sep 18 2013 *)
%o A133294 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-2*x-10*x^2)) \\ _G. C. Greubel_, Aug 02 2019
%o A133294 (Magma) I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +10*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Aug 02 2019
%o A133294 (Sage) ((1-x)/(1-2*x-10*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 02 2019
%o A133294 (GAP) a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+10*a[n-2]; od; a; # _G. C. Greubel_, Aug 02 2019
%Y A133294 Cf. A083101, A090042.
%K A133294 nonn,easy
%O A133294 0,3
%A A133294 _Philippe Deléham_, Dec 20 2007
%E A133294 Terms a(23) onward added by _G. C. Greubel_, Aug 02 2019