A116573 - OEIS

%I #7 Sep 04 2013 02:21:24

%S 1,0,4,17,1,82,324,49,961,5185,2501,5776,57600,54290,15625,497026,

%T 801025,1,3437317,9120400,1256641,18714277,85766122,38850289,72999937

%N A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as its first derivative InverseZtransform: A000073.

%C A polynomial derived in Mathematica by Bob Hanlon that is different from that in A000073; the first derivative sequence is different as well. Bob Hanlon's code: Needs["DiscreteMath`RSolve`"]; eqns={a[n]==a[n-1]+a[n-2]+a[n-3], a[0]==0,a[1]==a[2]==1}; Clear[f0,f1,f2,f3]; f0[0]=0;f0[1]=f0[2]=1; f0[n_Integer?Positive]:= f0[n]=f0[n-1]+f0[n-2]+f0[n-3]; f1[n_Integer]=a[n]/. RSolve[eqns,a[n],n][[1]]// ToRadicals//Simplify; (*Note that f1[n] is not restricted to nonnegative values of n.*) (*RSolve can also provide the generating function*) gf[x_]=GeneratingFunction[ eqns,a[n],n,x][[1,1]] -(x/(x^3 + x^2 + x - 1)) f2[n_Integer?NonNegative]:= SeriesCoefficient[ Series[gf[x],{x,0,n}],n];

%D Private email from Bob Hanlon (hanlonr(AT)cox.net), Mar 18 2006

%F g[x_] = -(x/(x^3 + x^2 + x - 1)); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; a(n) =Abs[w[n]]^2

%t g[x_] = -(x/(x^3 + x^2 + x - 1)); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; Table[Abs[Floor[N[w[n]]]]^2, {n, 1, 25}]

%Y Cf. A000073.

%K nonn,uned,obsc

%O 0,3

%A _Roger L. Bagula_, Mar 19 2006