A074062 Reflected (see A074058) pentanacci numbers A074048.

5, -1, -1, -1, -1, 9, -7, -1, -1, -1, 19, -23, 5, -1, -1, 39, -65, 33, -7, -1, 79, -169, 131, -47, 5, 159, -417, 431, -225, 57, 313, -993, 1279, -881, 339, 569, -2299, 3551, -3041, 1559, 799, -5167, 9401, -9633, 6159, 39, -11133, 23969, -28667, 21951, -6081, -22305

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 17 2002

Keywords

• easy
• sign

Comments

a(n) is also the trace of A^(-n), where A is the matrix ( (1,1,0,0,0), (1,0,1,0,0), (1,0,0,1,0), (1,0,0,0,1), (1,0,0,0,0) ).

a(n) is also the sum of determinants of 4th-order principal minors of A^n.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,1).

Crossrefs

Cf. A061084, A074058, A074048.

Programs

Magma

I:=[5,-1,-1,-1,-1]; [n le 5 select I[n] else (-1)*(Self(n-1) +Self(n-2) +Self(n-3) +Self(n-4)) + Self(n-5): n in [1..61]]; // G. C. Greubel, Jul 05 2021

Mathematica

CoefficientList[Series[(5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5), {x, 0, 60}], x]

PARI

Vec((5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5) + O(x^60)) \\ Michel Marcus, Sep 14 2020

Sage

def A074062_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5) ).list()
A074062_list(60) # G. C. Greubel, Jul 05 2021

Formula

a(n) = -a(n-1) -a(n-2) -a(n-3) -a(n-4) +a(n-5), a(0)=5, a(1)=-1, a(2)=-1, a(3)=-1, a(4)=-1.

G.f.: (5 +4*x +3*x^2 +2*x^3 +x^4)/(1 +x +x^2 +x^3 +x^4 -x^5).

Extensions

More terms from Michel Marcus, Sep 14 2020