login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A123518 Number of dumbbells in all possible arrangements of dumbbells on a 2 X n rectangular array of compartments. 1
1, 8, 38, 166, 671, 2602, 9792, 36068, 130697, 467556, 1655406, 5811290, 20255279, 70172502, 241839184, 829685064, 2835099649, 9653650752, 32768012102, 110913651342, 374469646511, 1261386990850, 4240037471152, 14225209349036 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 26 05:15 EDT 2022. Contains 354877 sequences. (Running on oeis4.)