A255983 - OEIS

%I #34 Sep 08 2022 08:46:11

%S 1,1,1,1,1,1,-4914,-181854,-4339944,-89153184,-1746815574,

%T -33850986114,-655203251304,-12686085675144,-245683477042884,

%U -4758284508073524,-92156792465163564,-1784855834560787004,-34568319709081645344,-669504074781304567584,-12966661247726595160224

%N a(n) = 1 for n <= 5; a(n) = 36*a(n-1) - 450*a(n-2) + 3000*a(n-3) - 11250*a(n-4) + 22500*a(n-5) - 18750*a(n-6) otherwise.

%C a(n)/a(n-1) tends to 19.367561... = 6 + 6^(1/6) + 6^(2/6) + 6^(3/6) + 6^(4/6) + 6^(5/6), the largest real root of the polynomial x^6 - 36*x^5 + 450*x^4 - 3000*x^3 + 11250*x^2 - 22500*x + 18750.

%C In general, the polynomial x^6 - k6*x^5 - k5*x^4 - k4*x^3 - k3*x^2 - k2*x -k1 has a root r+b*m^(1/6)+c*m^(2/6)+d*m^(3/6)+g*m^(4/6)+h*m^(5/6), see links for coefficients k1, k2, k3, k4, k5, k6.

%H Alexander Samokrutov, <a href="/A255983/b255983.txt">Table of n, a(n) for n = 0..25</a>

%H Alexander Samokrutov, <a href="/A255983/a255983_1.txt">Coefficients k1, k2, k3, k4, k5, k6</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (36,-450,3000,-11250,22500,-18750).

%F a(n) = 36*a(n-1) - 450*a(n-2) + 3000*a(n-3) - 11250*a(n-4) + 22500*a(n-5) - 18750*a(n-6).

%F G.f.: -(13835*x^5-8665*x^4+2585*x^3-415*x^2+35*x-1) / (18750*x^6-22500*x^5+11250*x^4-3000*x^3+450*x^2-36*x+1). - _Colin Barker_, Mar 23 2015

%t LinearRecurrence[{36, -450, 3000, -11250, 22500, -18750}, {1, 1, 1, 1, 1, 1}, 30] (* _Vincenzo Librandi_, Mar 21 2015 *)

%o (Magma) [n le 6 select 1 else 36*Self(n-1)-450*Self(n-2)+3000*Self(n-3)-11250*Self(n-4)+22500*Self(n-5)-18750*Self(n-6): n in [1..30]]; // _Vincenzo Librandi_, Mar 21 2015

%o (PARI) Vec(-(13835*x^5-8665*x^4+2585*x^3-415*x^2+35*x-1) / (18750*x^6-22500*x^5+11250*x^4-3000*x^3+450*x^2-36*x+1) + O(x^100)) \\ _Colin Barker_, Mar 23 2015

%Y Cf. A247344, A255985.

%K sign,easy

%O 0,7

%A _Alexander Samokrutov_, Mar 13 2015