[Fwd: new sequences derived in Mathematica this morning from Bob Hanlon's polynomial]

*To*: mathgroup at smc.vnet.net*Subject*: [mg65736] [Fwd: new sequences derived in Mathematica this morning from Bob Hanlon's polynomial]*From*: Roger Bagula <rlbagulatftn at yahoo.com>*Date*: Sun, 16 Apr 2006 03:49:15 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

[This post has been delayed due to email problems - moderator] This result of Bob Hanlon's may be historical: the machinne using Mathematica beats the expert Mathematican at a derivation! http://www.research.att.com/~njas/sequences/?q=A000073&language=english&go=Search -------- Original Message -------- Subject: [mg65736] new sequences derived in Mathematica this morning from Bob Hanlon's polynomial From: Roger Bagula <rlbagulatftn at yahoo.com> To: mathgroup at smc.vnet.net Bob Hanlon said I could use his code, so I submitted these two sequences. Does the machine beat the Mathematician? People have long talked about machine derived mathematics. We are actually doing that with this Binet type mathematics! %I A000001 %S A000001 0, 1, 10, 1, 49, 225, 36, 730, 4097, 2025, 4761, 48401, 46225, 13456, 432965, 703922, 1, 3066002, 8185321, 1134225, 16974401, 78145601, 35545444, 67043345, 632572802 %N A000001 A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as it first derivative InverseZtransform: A000073 %C A000001 x^2/(1 - x - x^2 - x^3) is similar to the polynomial: -(x/(x^3 + x^2 + x - 1)) but not the same. As the last is machine derived , it is probly more correct than the one quoted presently in A000072. %F A000001 (*Source : A000073*) g[x_] = x^2/(1 - x - x^2 - x^3); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; a(n) =Abs[w[n]]^2 %t A000001 (*Source : A000073*) g[x_] = x^2/(1 - x - x^2 - x^3); 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 A000001 Cf. A000073 %O A000001 0 %K A000001 ,nonn, %A A000001 Roger L. Bagula (rlbagulatftn at yahoo.com), Mar 19 2006 RH RA 69.225.126.84 RU RI %I A000001 %S A000001 1, 0, 4, 17, 1, 82, 324, 49, 961, 5185, 2501, 5776, 57600, 54290, 15625, 497026, 801025, 1, 3437317, 9120400, 1256641, 18714277, 85766122, 38850289, 72999937 %N A000001 A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as it first derivative InverseZtransform: A000073 %C A000001 A polynomial derived in Mathematica by bob hanlon that is different than that in A000073 : 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/(x3 + x2 + x - 1)) f2[n_Integer?NonNegative]:= SeriesCoefficient[ Series[gf[x],{x,0,n}],n]; %D A000001 Private email from Bob Hanlon: From: Bob Hanlon <hanlonr at cox.net> To: mathgroup at smc.vnet.net > Date: 2006/03/18 Sat PM 05:20:38 EST %F A000001 g[x_] = -(x/(x3 + x2 + x - 1)); dg[x_] = D[g[x], {x, 1}]; w[n_] := InverseZTransform[dg[x], x, n] // ToRadicals; a(n) =Abs[w[n]]^2 %t A000001 g[x_] = -(x/(x3 + x2 + 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 A000001 Cf. A000073 %O A000001 0 %K A000001 ,nonn, %A A000001 Roger L. Bagula (rlbagulatftn at yahoo.com), Mar 19 2006 RH RA 69.225.126.84 RU RI Roger L. Bagula { email: rlbagula at sbcglobal.net or rlbagulatftn at yahoo.com } 11759 Waterhill Road, Lakeside, Ca. 92040 telephone: 619-561-0814