A376080 a(n) is the highest degree of the rational function in the recursive substitution {y1, y2} -> {y2, (y2 + 1)/(y1*y2)} after n steps.

1, 1, 1, 2, 4, 5, 7, 10, 13, 16, 20, 25, 29, 34, 40, 46, 52, 59, 67, 74, 82, 91, 100, 109, 119, 130, 140, 151, 163, 175, 187, 200, 214, 227, 241, 256, 271, 286, 302, 319, 335, 352, 370, 388, 406, 425, 445, 464, 484, 505, 526, 547, 569, 592, 614, 637, 661, 685, 709, 734, 760, 785, 811, 838, 865

Offset: 0

Author

Thomas Scheuerle, Sep 09 2024

Keywords

• nonn
• easy

Comments

An example where the degree of the n-th iterate of a rational map exhibits polynomial growth. Also an example for exponential growth was given in the thesis from Khaled Hamad by A011782.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Khaled Hamad, Laurentification, Thesis (2017). La Trobe University.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Crossrefs

Cf. A011858, A058937.

Cf. A011782 (highest degree of the rational function in the substitution: {y1, y2} -> {y2, y2 + y1/y2}).

Programs

Mathematica

A376080[n_] := Ceiling[(3*n*(n - 1) + 8)/14];
Array[A376080, 100, 0] (* Paolo Xausa, Sep 23 2024 *)

PARI

r(v) = [v[2], (v[2]+1)/(v[1]*v[2])];
a(n) = {my(v = [x,x]); if(n < 2, 1,for(k=0, n-2, v = r(v)); poldegree(numerator(v[2])))};

Formula

G.f.: (1 - x + x^3 + x^4 - x^5 + x^6 + x^8)/(1 - 2*x + x^2 - x^7 + 2*x^8 - x^9).

a(n) = ceiling((3*n^2 - 3*n + 8)/14).

a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9).

a(n) = a(n-7) + 3*(n-7) + 9.

(2*a(n+6) - a(n+5) - 2*a(n-1) + a(n-2) - 9)/3 = n.