A083425 a(n) = (5*5^n + (-1)^n)/6.

1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021, 15894571940104, 79472859700521, 397364298502604, 1986821492513021, 9934107462565104

Offset: 0

Author

Paul Barry, Apr 30 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A083424. Inverse binomial transform of A052934.

Primes occur at indices n = 4, 66, 100, 102, 228, 346, ..., see A138647. - R. J. Mathar, Jan 19 2011

Sum_{i=0..m} (-1)^(m+i)*5^i, for m >= 0, gives all terms of the sequence. - Bruno Berselli, Aug 28 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,5).

Programs

GAP

List([0..25],n->(5*5^n+(-1)^n)/6); # Muniru A Asiru, Sep 21 2018

Magma

[n le 2 select n^2 else 4*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012

Maple

seq(coeff(series(factorial(n)*(5*exp(5*x)+exp(-x))/6,x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Sep 21 2018

Mathematica

LinearRecurrence[{4,5},{1,4},40] (* Vincenzo Librandi, Jun 23 2012 *)

PARI

a(n)=(5*5^n+(-1)^n)/6 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (5*5^n + (-1)^n)/6.

G.f.: 1/((1+x)*(1-5x)).

E.g.f.: (5*exp(5x) + exp(-x))/6.

a(n) = Sum_{k=0..n} C(n-k,k)*4^(n-2k)*5^k. - Paul Barry, Jul 29 2004

a(n) = A015531(n+1). - R. J. Mathar, Sep 17 2008

a(n) = 4*a(n-1) + 5*a(n-2). - Vincenzo Librandi, Jun 23 2012