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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, 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
0,4
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
Khaled Hamad, Laurentification, Thesis (2017). La Trobe University.
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.
MATHEMATICA
A376080[n_] := Ceiling[(3*n*(n - 1) + 8)/14];
Array[A376080, 100, 0] (* Paolo Xausa, Sep 23 2024 *)
PROG
(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])))};
CROSSREFS
Cf. A011782 (highest degree of the rational function in the substitution: {y1, y2} -> {y2, y2 + y1/y2}).
Sequence in context: A036559 A083022 A279022 * A226807 A211523 A340246
KEYWORD
nonn,easy
AUTHOR
Thomas Scheuerle, Sep 09 2024
STATUS
approved