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

1,2

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).

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.

Example

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.

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 *)

Prog

(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
(Magma) R<x>:=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
(Sage)
def A123518_list(prec):
    P.<x> = 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
(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

Crossrefs

Cf. A030186, A046741.

Sequence in context: A026640 A122682 A225429 * A197338 A214931 A229366

Adjacent sequences: A123515 A123516 A123517 * A123519 A123520 A123521

Keyword

nonn

Author

Emeric Deutsch, Oct 16 2006

Status

approved