A050401 Number of independent sets of nodes in P_4 X C_n (n > 2).

8, 1, 41, 142, 933, 4741, 26660, 143697, 788453, 4293286, 23454801, 127953981, 698467368, 3811712633, 20803963753, 113540081302, 619672701957, 3381980484909, 18457878595412, 100737602247769, 549796303339413

Offset: 0

Author

Stephen G Penrice, Dec 21 1999

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,20,27,-14,-25,4,5,-1).

Crossrefs

Column 4 of A286513.

Programs

GAP

a:=[8,1,41,142,933,4741,26660,143697];; for n in [9..30] do a[n]:= a[n-1]+20*a[n-2]+27*a[n-3]-14*a[n-4]-25*a[n-5]+4*a[n-6]+5*a[n-7]-a[n-8]; od; a; # G. C. Greubel, Oct 30 2019

Magma

I:=[8,1,41,142,933,4741,26660,143697]; [n le 8 select I[n] else Self(n-1)+20*Self(n-2)+27*Self(n-3)-14*Self(n-4)- 25*Self(n-5)+4*Self(n-6)+5*Self(n-7)-Self(n-8): n in [1..30]]; // Vincenzo Librandi, May 11 2017

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*( 1-4*x-9*x^2+5*x^3+4*x^4-x^5)) )); // G. C. Greubel, Oct 30 2019

Maple

seq(coeff(series((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/( (1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)), x, n+1), x, n), n = 0 ..30); # G. C. Greubel, Oct 30 2019

Mathematica

CoefficientList[Series[(8-7*x-120*x^2-135*x^3+56*x^4+75*x^5-8*x^6-5*x^7) /( (1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)), {x, 0, 50}], x] (* Vincenzo Librandi, May 11 2017 *)

PARI

my(x='x+O('x^30)); Vec((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5))) \\ G. C. Greubel, Oct 30 2019

Sage

def A050401_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5))).list()
A050401_list(30) # G. C. Greubel, Oct 30 2019

Formula

a(n) = a(n-1) + 20*a(n-2) + 27*a(n-3) - 14*a(n-4) - 25*a(n-5) + 4*a(n-6) + 5*a(n-7) - a(n-8).

G.f.: (8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)). - Colin Barker, Aug 31 2012

Extensions

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999