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%I #44 Feb 18 2024 08:27:14
%S 1,2,10,38,154,614,2458,9830,39322,157286,629146,2516582,10066330,
%T 40265318,161061274,644245094,2576980378,10307921510,41231686042,
%U 164926744166,659706976666,2638827906662,10555311626650,42221246506598
%N a(n) = 3*a(n-1) + 4*a(n-2), with a(0)=1, a(1)=2.
%C Inverse binomial transform of A005053. Binomial transform of [1, 1, 7, 13, 55, ...] = A015441(n+1).
%C Convolved with [1, 2, 2, 2, ...] = powers of 4: [1, 4, 16, 64, ...]. - _Gary W. Adamson_, Jun 02 2009
%C a(n) is the number of compositions of n when there are 2 types of 1 and 6 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%H Vincenzo Librandi, <a href="/A122117/b122117.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 (3,4).
%F a(n) = 2*A108981(n-1) for n > 0, with a(0) = 1.
%F a(2*n) = 4*a(2*n-1) + 2, a(2*n+1) = 4*a(2*n) - 2.
%F a(n) = Sum_{k=0..n} 2^(n-k)*A055380(n,k).
%F G.f.: (1-x)/(1-3*x-4*x^2).
%F Lim_{n->infinity} a(n+1)/a(n) = 4.
%F a(n) = Sum_{k=0..n} A122016(n,k)*2^k. - _Philippe Deléham_, Nov 05 2008
%t CoefficientList[Series[(1-x)/(1-3*x-4*x^2),{x,0,30}],x] (* _Vincenzo Librandi_, Jul 06 2012 *)
%o (Sage) from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,3,4, lambda n: 0); [next(it) for i in range(24)] # _Zerinvary Lajos_, Jul 03 2008
%o (Sage) ((1-x)/(1-3*x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 18 2019
%o (PARI) Vec((1-x)/(1-3*x-4*x^2)+O(x^30)) \\ _Charles R Greathouse IV_, Jan 11 2012
%o (Magma) I:=[1, 2]; [n le 2 select I[n] else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jul 06 2012
%o (GAP) a:=[1,2];; for n in [3..30] do a[n]:=3*a[n-1]+4*a[n-2]; od; a; # _G. C. Greubel_, May 18 2019
%Y Cf. A201455.
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Oct 19 2006
%E Corrected by _T. D. Noe_, Nov 07 2006
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