|
%I #31 Sep 08 2022 08:45:55
%S 1,1,1,1,3,5,9,13,17,23,29,37,45,53,63,73,85,97,109,123,137,153,169,
%T 185,203,221,241,261,281,303,325,349,373,397,423,449,477,505,533,563,
%U 593,625,657,689,723,757,793,829,865,903,941,981,1021,1061,1103,1145
%N a(n) = floor( (2*n^2 - 6*n + 9) / 5).
%C Hasselblatt and Propp on page 8 mentions the sequence as degrees of iterates of (w, x, y, z) -> (x, y, z, z*(w*z - x*y) / (w*y - x*x)). That is, if b(0) = w, b(1) = x, b(2) = y, b(3) = z, b(n) = b(n-1) * (b(n-1)*b(n-4) - b(n-2)*b(n-3)) / (b(n-2)*b(n-4) - b(n-3)*b(n-3)), then b(n) is a rational function such that the total degree of the numerator is a(n) and the denominator is a(n)-1. Also b(n) is a Laurent monomial in variables {w, x, y, z, wz-xy, wy-xx, xz-yy}.
%C A quasipolynomial. - _Charles R Greathouse IV_, Dec 28 2012
%H Harvey P. Dale, <a href="/A185170/b185170.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Hasselblatt and J. Propp, <a href="http://arxiv.org/abs/math/0604521">Degree-Growth of Monomial Maps</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 1, -2, 1).
%F G.f.: (1 - x + 2*x^4 - x^5 + 3*x^6) / ((1 - x)^2 * (1 - x^5)).
%F a(n) = a(3-n) for all n in Z.
%F Second difference has period 5.
%F a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=3, a(5)=5, a(6)=9, a(n)=2*a(n-1)- a(n-2)+ a(n-5)-2*a(n-6)+a (n-7). - _Harvey P. Dale_, Dec 28 2012
%e G.f. = 1 + x + x^2 + x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 13*x^7 + 17*x^8 + 23*x^9 + ...
%t Table[Floor[(2 n^2 - 6 n + 9)/5], {n, 0, 60}] (* or *) LinearRecurrence[ {2,-1,0,0,1,-2,1},{1,1,1,1,3,5,9},60] (* _Harvey P. Dale_, Dec 28 2012 *)
%t a[ n_] := Quotient[ 2 n^2 - 6 n + 9, 5]; (* _Michael Somos_, Apr 25 2015 *)
%o (PARI) {a(n) = (2*n^2 - 6*n + 9) \ 5};
%o (Maxima) A185170(n):=floor((2*n^2-6*n+9)/5)$ makelist(A185170(n),n,0,30); /* _Martin Ettl_, Dec 28 2012 */
%o (Magma) [Floor((2*n^2-6*n+9)/5): n in [0..50]]; // _G. C. Greubel_, Aug 10 2018
%Y Cf. A130520.
%K nonn
%O 0,5
%A _Michael Somos_, Dec 26 2012
|
