A127212 a(n) = 5^n*Lucas(n), where Lucas = A000204.

5, 75, 500, 4375, 34375, 281250, 2265625, 18359375, 148437500, 1201171875, 9716796875, 78613281250, 635986328125, 5145263671875, 41625976562500, 336761474609375, 2724456787109375, 22041320800781250, 178318023681640625

Offset: 1

Author

Artur Jasinski, Jan 09 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,25).

Crossrefs

Cf. A000204, A087131, A127210, A127211, A127213, A127214, A127215, A127216.

Programs

Magma

[5^n*Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 18 2017

Mathematica

Table[5^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
Table[5^n*LucasL[n], {n,1,50}] (* G. C. Greubel, Dec 18 2017 *)
LinearRecurrence[{5,25},{5,75},20] (* Harvey P. Dale, Jan 11 2024 *)

PARI

x='x+O('x^30); Vec(-5*x*(10*x+1)/(25*x^2+5*x-1)) \\ G. C. Greubel, Dec 18 2017

Formula

a(n) = Trace of matrix [({5,5},{5,0})^n].

a(n) = 5^n * Trace of matrix [({1,1},{1,0})^n].

From Colin Barker, Sep 02 2013: (Start)

a(n) = 5*a(n-1) + 25*a(n-2).

G.f.: -5*x*(10*x+1)/(25*x^2+5*x-1). (End)