A247344 a(n) = 1 for n <= 4; a(n) = 25*a(n-1) - 200*a(n-2) + 800*a(n-3) - 1600*a(n-4) + 1280*a(n-5) otherwise.

1, 1, 1, 1, 1, 305, 7905, 137105, 2090305, 30673905, 446213025, 6483539025, 94216001025, 1369259983025, 19900452349025, 289229603172625, 4203610924242625, 61094494859232625, 887935798190222625

Offset: 0

Author

Alexander Samokrutov, Sep 14 2014

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to 14.5338... = 5 + 5^(1/5)+5^(2/5)+5^(3/5)+5^(4/5) = 4/(1-5^(-1/5)), the real root of the polynomial x^5 - 25*x^4 + 200*x^3 - 800*x^2 + 1600*x - 1280.

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

Links

• Alexander Samokrutov, Table of n, a(n) for n = 0..22
• Alexander Samokrutov, Coefficients k1, k2, k3, k4, k5
• Index entries for linear recurrences with constant coefficients, signature (25,-200,800,-1600,1280).

Crossrefs

Cf. A255985, A255983.

Programs

Magma

[n le 5 select 1 else 25*Self(n-1)-200*Self(n-2)+800*Self(n-3)-1600*Self(n-4)+1280*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Nov 19 2014

Mathematica

CoefficientList[Series[(976 x^4 - 624 x^3 + 176 x^2 - 24 x + 1) / (-1280 x^5 + 1600 x^4 - 800 x^3 + 200 x^2 - 25 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 19 2014 *)

PARI

Vec( (976*x^4 - 624*x^3 + 176*x^2 - 24*x + 1)/(-1280*x^5 + 1600*x^4 - 800*x^3 + 200*x^2 - 25*x + 1) + O(x^66) ) \\ Joerg Arndt, Sep 14 2014

Formula

a(n) = 25*a(n-1)-200*a(n-2)+800*a(n-3)-1600*a(n-4)+1280*a(n-5).

G.f.: (976*x^4 - 624*x^3 + 176*x^2 - 24*x + 1)/(-1280*x^5 + 1600*x^4 - 800*x^3 + 200*x^2 - 25*x + 1).