A123518 Number of dumbbells in all possible arrangements of dumbbells on a 2 X n rectangular array of compartments.

1, 8, 38, 166, 671, 2602, 9792, 36068, 130697, 467556, 1655406, 5811290, 20255279, 70172502, 241839184, 829685064, 2835099649, 9653650752, 32768012102, 110913651342, 374469646511, 1261386990850, 4240037471152, 14225209349036

Offset: 1

Author

Emeric Deutsch, Oct 16 2006

Keywords

• nonn

Examples

			a(2)=8 because in a 2 X 2 array of compartments, numbered clockwise starting from the NW one, we have 7 (=A030186(2)) possible arrangements of dumbbells: [ ], [14], [23], [12], [34], [14,23] and [12,34] (ij indicates a dumbbell placed in the compartments i and j); these contain altogether 8 dumbbells.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
• R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
• R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
• Index entries for linear recurrences with constant coefficients, signature (6,-7,-8,5,2,-1).

Crossrefs

Cf. A030186, A046741.

Programs

GAP

a:=[1,8,38,166,671,2602];; for n in [7..30] do a[n]:=6*a[n-1] -7*a[n-2]-8*a[n-3]+5*a[n-4]+2*a[n-5]-a[n-6]; od; a; # G. C. Greubel, Oct 28 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2 )); // G. C. Greubel, Oct 28 2019

Maple

G:=z*(1+2*z-3*z^2+2*z^3)/(1-3*z-z^2+z^3)^2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..27);

Mathematica

LinearRecurrence[{6,-7,-8,5,2,-1}, {1,8,38,166,671,2602}, 30] (* G. C. Greubel, Oct 28 2019 *)

PARI

my(x='x+O('x^30)); Vec(x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2) \\ G. C. Greubel, Oct 28 2019

Sage

def A123518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2 ).list()
a=A123518_list(30); a[1:] # G. C. Greubel, Oct 28 2019

Formula

a(n) = Sum_{k=0..n} k*A046741(n,k).

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