A097600 A Binet like formula using the Akiyama-Thurston tile roots for a Minimal Pisot theta0 sequence.

1, 0, 1, 2, 2, 3, 4, 5, 7, 10, 13, 18, 23, 31, 41, 55, 73, 97, 129, 170, 226, 299, 397, 526, 696, 923, 1223, 1620, 2146, 2843, 3766, 4989, 6610, 8756, 11599, 15366, 20356, 26966, 35723, 47323, 62689, 83046, 110013, 145736, 193059, 255749, 338796

Offset: 1

Author

Roger L. Bagula, Sep 20 2004

Keywords

• nonn
• uned
• less

Comments

Let r1 = -0.662358978622373051..-0.562279512062301289..*i, r2 = complex-conjugate(r1), and r3 = 1.3247179572.. = A060006 be the three roots of the polynomial x^3-x-1. i is the imaginary unit. Then f(n) = (r3^n-r2^n-r2^(5*n))/(r3-r2-r2^5) is a sequence of numbers, approximately f(1) = 1, f(2) = 0.756+0.786*i, f(3) = 1.263+0.017*i, f(4) = 2.1929+0.704*i, f(5) = 2.205+0.6866*i etc. a(n) is floor(Re(f(n)).

Links

• Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

Crossrefs

Cf. A001644.

Programs

Mathematica

NSolve[x^3-x-1==0, x] r1=-0.662358978622373051`-0.562279512062301289` I r2=-0.662358978622373051`+0.562279512062301289` I r3=1.32471795724474605` (* Binet like formula for the Minimal Pisot*) f[n_]=(r3^n-((r2^n)+(r2^(5*n))))/(r3-r2-r2^5) a=Table[Floor[Re[f[n]]], {n, 1, 50}]