A014987 a(n) = (1 - (-6)^n)/7.

1, -5, 31, -185, 1111, -6665, 39991, -239945, 1439671, -8638025, 51828151, -310968905, 1865813431, -11194880585, 67169283511, -403015701065, 2418094206391, -14508565238345, 87051391430071, -522308348580425, 3133850091482551

Offset: 1

Author

Olivier Gérard

Keywords

• sign
• easy

Comments

q-integers for q=-6.

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-1)=(-1)^n*charpoly(A,2). - Milan Janjic, Jan 27 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (-5,6).

Crossrefs

Absolute values are in A015540.

Cf. A077925, A014983, A014985, A014986, A014989-A014994.

Programs

Magma

I:=[1,-5]; [n le 2 select I[n] else -5*Self(n-1)+6*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 22 2012

Maple

a:=n->sum ((-6)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

Mathematica

LinearRecurrence[{-5, 6}, {1, -5}, 30] (* Vincenzo Librandi Oct 22 2012 *)

PARI

a(n)=(1-(-6)^n)/7 \\ Charles R Greathouse IV, Sep 24 2015

Sage

[gaussian_binomial(n,1,-6) 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+6*x)*(1-x)).

a(n) = -5*a(n-1) + 6*a(n-2). - Vincenzo Librandi Oct 22 2012

E.g.f.: (exp(x) - exp(-6*x))/7. - G. C. Greubel, May 26 2018

Extensions

Better name from Ralf Stephan, Jul 14 2013