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A116573 A Binet type formula from a polynomial whose coefficient expansion gives a tribonacci used as its first derivative InverseZtransform: A000073. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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];

REFERENCES

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

LINKS

Table of n, a(n) for n=0..24.

FORMULA

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

MATHEMATICA

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}]

CROSSREFS

Cf. A000073.

Sequence in context: A296628 A123234 A198297 * A195881 A264217 A341442

Adjacent sequences:  A116570 A116571 A116572 * A116574 A116575 A116576

KEYWORD

nonn,uned,obsc

AUTHOR

Roger L. Bagula, Mar 19 2006

STATUS

approved

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Last modified September 24 03:26 EDT 2021. Contains 347623 sequences. (Running on oeis4.)