A055608 Arrays of dumbbells.

1, 13, 92, 473, 1982, 7191, 23431, 70234, 196941, 522939, 1327002, 3240917, 7660538, 17602967, 39466363, 86593478, 186399956, 394478234, 822229746, 1690521204, 3433033150, 6893852746, 13702694284, 26982983126, 52680389239

Offset: 1

Author

Henry Bottomley, Jun 02 2000

Keywords

• easy
• nonn

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.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-31,44,4,-84,66,46,-74,-4,36,-4,-9,1,1).

Crossrefs

Cf. A002940, A002941, A002889, A046741.

Cf. A062123-A062127.

Programs

Haskell

a055608 n = a055608_list !! (n-1)
a055608_list = 1 : 13 : 92 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a055608_list) a055608_list)
   (drop 2 $ zipWith (+) (tail a002889_list) a002889_list)
-- Reinhard Zumkeller, Jan 18 2014

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^4/((1-x)^4*(1-x-x^2)^5) )); // G. C. Greubel, Jan 31 2019

Mathematica

CoefficientList[Series[(1+x)^4/((1-x)^4*(1-x-x^2)^5), {x,0,30}], x] (* G. C. Greubel, Jan 31 2019 *)

PARI

my(x='x+O('x^30)); Vec((1+x)^4/((1-x)^4*(1-x-x^2)^5)) \\ G. C. Greubel, Jan 31 2019

Sage

((1+x)^4/((1-x)^4*(1-x-x^2)^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

Formula

G.f.: (1+x)^4/((1-x)^4*(1-x-x^2)^5).

a(n) = 2*a(n-1) - a(n-3) + A002889(n) + A002889(n-1).