A014986 a(n) = (1 - (-5)^n)/6.

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

Offset: 1

Author

Olivier Gérard

Keywords

• sign
• easy

Comments

q-integers for q = -5.

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 27 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (-4,5).

Crossrefs

Cf. A077925, A014983, A014985, A014987, A014989, A014990, A014991, A014992, A014993, A014994.

Programs

Magma

I:=[1, -4]; [n le 2 select I[n] else -4*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2012

Maple

a:=n->sum ((-5)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

Mathematica

LinearRecurrence[{-4,5},{1,-4},30] (* Vincenzo Librandi, Jun 19 2012 *)

PARI

a(n)=(1-(-5)^n)/6 \\ Charles R Greathouse IV, Dec 07 2011

Sage

[gaussian_binomial(n,1,-5) for n in range(1,22)] # Zerinvary Lajos, May 28 2009

Formula

a(n) = a(n-1) + q^(n-1) = (q^n - 1) / (q - 1).

G.f.: x/((1-x)*(1+5*x)). - Bruno Berselli, Dec 07 2011

a(n) = -4*a(n-1) + 5*a(n-2). - Vincenzo Librandi, Jun 19 2012

E.g.f.: (exp(x) - exp(-5*x))/6. - G. C. Greubel, May 26 2018

Extensions

Better name from Ralf Stephan, Jul 14 2013