A376080 - OEIS

%I #34 Sep 23 2024 11:32:15

%S 1,1,1,2,4,5,7,10,13,16,20,25,29,34,40,46,52,59,67,74,82,91,100,109,

%T 119,130,140,151,163,175,187,200,214,227,241,256,271,286,302,319,335,

%U 352,370,388,406,425,445,464,484,505,526,547,569,592,614,637,661,685,709,734,760,785,811,838,865

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

%C 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.

%H Paolo Xausa, <a href="/A376080/b376080.txt">Table of n, a(n) for n = 0..10000</a>

%H Khaled Hamad, <a href="https://web.archive.org/web/20230121195132id_/https://ltu-figshare-repo.s3.aarnet.edu.au/ltu-figshare-repo/38787108/42508_SOURCE01_2_A.pdf?AWSAccessKeyId=RADjuIEnIStOwNiA&amp;Expires=1674330702&amp;Signature=dI44kcVYhLdlAlTuO%2Ffq5ANYhpk%3D">Laurentification</a>, Thesis (2017). La Trobe University.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,1,-2,1).

%F 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).

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

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

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

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

%t A376080[n_] := Ceiling[(3*n*(n - 1) + 8)/14];

%t Array[A376080, 100, 0] (* _Paolo Xausa_, Sep 23 2024 *)

%o (PARI)

%o r(v) = [v[2], (v[2]+1)/(v[1]*v[2])];

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

%Y Cf. A011858, A058937.

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

%K nonn,easy

%O 0,4

%A _Thomas Scheuerle_, Sep 09 2024