%I #33 Apr 08 2021 03:41:37

%S 1,4,11,34,101,304,911,2734,8201,24604,73811,221434,664301,1992904,

%T 5978711,17936134,53808401,161425204,484275611,1452826834,4358480501,

%U 13075441504,39226324511,117678973534,353036920601

%N a(n) = 2*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 4.

%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=charpoly(A,2). - _Milan Janjic_, Jan 26 2010

%H Harry J. Smith, <a href="/A060925/b060925.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,3).

%F Row sums of Lucas convolution triangle A060922.

%F Inverse binomial transform of A003947. - _Philippe Deléham_, Jul 23 2005

%F a(n) = Sum_{m=0..n} A060922(n, m) = Sum_{j=1..n} (a(j-1)*A000204(n-j+1)) + A000204(n+1).

%F a(n) = (5*3^n - (-1)^n)/4.

%F G.f.: (1+2*x)/(1 - 2*x - 3*x^2).

%F a(2n) = 3*a(2n-1) - 1; a(2n+1) = 3*a(2n) + 1. - _Philippe Deléham_, Jul 23 2005

%F Binomial transform is A003947. - _Paul Barry_, May 19 2003

%F E.g.f.: (-exp(-x) + 5*exp(3*x))/4. - _G. C. Greubel_, Apr 06 2021

%p A060925:= n-> (5*3^n - (-1)^n)/4; seq(A060925(n), n=0..30); # _G. C. Greubel_, Apr 06 2021

%t f[n_]:=3/(n+2);x=2;Table[x=f[x];Denominator[x],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2010 *)

%t LinearRecurrence[{2,3},{1,4},30] (* _Harvey P. Dale_, Mar 07 2014 *)

%o (PARI) {a(n) = (5*3^n - (-1)^n)/4};

%o vector(30, n, a(n-1)) \\ _Harry J. Smith_, Jul 19 2009 \\ modified by _G. C. Greubel_, Apr 06 2021

%o (Magma) [(5*3^n - (-1)^n)/4: n in [0..30]]; // _G. C. Greubel_, Apr 06 2021

%o (Sage) [(5*3^n - (-1)^n)/4 for n in (0..30)] # _G. C. Greubel_, Apr 06 2021

%Y Cf. A000204, A003947, A060922.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Apr 20 2001

%E Recurrence, now used as definition, from _Philippe Deléham_, Jul 23 2005

%E Entry revised by _N. J. A. Sloane_, Sep 10 2006