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

1,4

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

Table of n, a(n) for n=1..47.

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.

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}]

Crossrefs

Cf. A001644.

Sequence in context: A282949 A292200 A249576 * A136422 A173674 A018128

Adjacent sequences: A097597 A097598 A097599 * A097601 A097602 A097603

Keyword

nonn,uned,less

Author

Roger L. Bagula, Sep 20 2004

Status

approved