A002889 Arrays of dumbbells.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A046741, A002940, A002941.

Cf. A055608, A062123-A062127.

Programs

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

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^3/((1-x)^3*(1-x-x^2)^4) )); // G. C. Greubel, Jan 31 2019

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 *)

PARI

x='x+O('x^30); Vec((1+x)^3/((1-x)^3*(1-x-x^2)^4)) \\ Altug Alkan, Jul 31 2018

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

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).

Extensions

More terms from Henry Bottomley, Jun 02 2000