A002889 Arrays of dumbbells. (Formerly M4715 N2016)

1, 10, 56, 234, 815, 2504, 7018, 18336, 45328, 107160, 244198, 539656, 1161987, 2446906, 5054440, 10266850, 20549117, 40595568, 79271188, 153190480, 293278496, 556737696, 1048772300, 1961855408, 3646420325, 6737649754

Offset

1,2

References

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, 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)

Index entries for linear recurrences with constant coefficients, signature (7,-17,11,19,-29,-3,21,-3,-7,1,1).

Formula

a(n) = 2*a(n-1) - a(n-3) + A002941(n) + A002941(n-1).

G.f.: (1+x)^3/((1-x)^3*(1-x-x^2)^4).

Mathematica

CoefficientList[(1+x)^3/((1-x)^3*(1-x-x^2)^4) + O[x]^30, x] (* Jean-François Alcover, Jul 31 2018 *)
LinearRecurrence[{7, -17, 11, 19, -29, -3, 21, -3, -7, 1, 1}, {1, 10, 56, 234, 815, 2504, 7018, 18336, 45328, 107160, 244198}, 30] (* Harvey P. Dale, Jul 25 2021 *)

Prog

(Haskell)
a002889 n = a002889_list !! (n-1)
a002889_list = 1 : 10 : 56 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a002889_list) a002889_list)
   (drop 2 $ zipWith (+) (tail a002941_list) a002941_list)
-- Reinhard Zumkeller, Jan 18 2014
(PARI) x='x+O('x^30); Vec((1+x)^3/((1-x)^3*(1-x-x^2)^4)) \\ Altug Alkan, Jul 31 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^3/((1-x)^3*(1-x-x^2)^4) )); // G. C. Greubel, Jan 31 2019
(Sage) ((1+x)^3/((1-x)^3*(1-x-x^2)^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

Crossrefs

Cf. A046741, A002940, A002941.

Cf. A055608, A062123-A062127.

Sequence in context: A320756 A053309 A035040 * A055911 A087076 A014483

Adjacent sequences: A002886 A002887 A002888 * A002890 A002891 A002892

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Henry Bottomley, Jun 02 2000

Status

approved