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a(n) = 2*a(n-1) + 10*a(n-2), a(0)=1, a(1)=1.
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%I #35 Jan 01 2024 11:44:46

%S 1,1,12,34,188,716,3312,13784,60688,259216,1125312,4842784,20938688,

%T 90305216,389997312,1683046784,7266066688,31362601216,135385869312,

%U 584397750784,2522654194688,10889285897216,47005113741312

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

%C 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 A083101 is an essentially identical sequence (with a different start). - _N. J. A. Sloane_, Dec 31 2012

%H G. C. Greubel, <a href="/A133294/b133294.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,10).

%F a(n) = Sum_{k=0..n} A098158(n,k)*11^(n-k).

%F G.f.: (1-x)/(1-2*x-10*x^2).

%F a(n) = A083101(n-1) for n >= 1.

%F 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 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 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-2*x-10*x^2)) \\ _G. C. Greubel_, Aug 02 2019

%o (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 (Sage) ((1-x)/(1-2*x-10*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 02 2019

%o (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 Cf. A083101, A090042.

%K nonn,easy

%O 0,3

%A _Philippe Deléham_, Dec 20 2007

%E Terms a(23) onward added by _G. C. Greubel_, Aug 02 2019