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[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
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