A050402 Number of independent sets of nodes in C_4 X C_n (n > 2).

7, 1, 35, 121, 743, 3561, 18995, 96433, 500871, 2573905, 13292995, 68492073, 353290343, 1821383097, 9392360019, 48428332641, 249716406791, 1287608913057, 6639354593123, 34234612471001, 176524935990503, 910219628918665, 4693389213891699, 24200638961917201

Offset: 0

Author

Stephen G Penrice, Dec 21 1999

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8
• Eric Weisstein's World of Mathematics, Independent Vertex Set
• Eric Weisstein's World of Mathematics, Torus Grid Graph
• Index entries for linear recurrences with constant coefficients, signature (2,15,8,-7,-2,1).

Programs

GAP

a:=[7,1,35,121,743,3561];; for n in [7..30] do a[n]:=2*a[n-1] +15*a[n-2]+8*a[n-3]-7*a[n-4]-2*a[n-5]-a[n-6]; od; a; # G. C. Greubel, Oct 30 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3)) )); // G. C. Greubel, Oct 30 2019

Maple

seq(coeff(series((7-13*x-72*x^2-20*x^3+17*x^4+x^5)/((1+x)*(1+2*x-x^2) *(1-5*x-x^2+x^3)), x, n+1), x, n), n = 0 ..30); # G. C. Greubel, Oct 30 2019

Mathematica

CoefficientList[Series[(7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3)), {x, 0, 30}], x]

PARI

Vec((7-13*x-72*x^2-20*x^3+17*x^4+x^5)/((1+x)*(1+2*x-x^2)*(1-5*x- x^2+x^3)) + O(x^30)) \\ Colin Barker, May 11 2017

Sage

def A050402_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3))).list()
A050402_list(30) # G. C. Greubel, Oct 30 2019

Formula

a(n) = a(n-1) + 17*a(n-2) + 23*a(n-3) + a(n-4) - 9*a(n-5) - a(n-6) + a(n-7).

G.f.: (7 -13*x -72*x^2 -20*x^3 +17*x^4 +x^5)/((1+x)*(1+2*x-x^2)*(1-5*x-x^2+x^3)). - Colin Barker, Aug 31 2012

Extensions

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