A002941 Arrays of dumbbells.

1, 7, 29, 94, 263, 667, 1577, 3538, 7622, 15900, 32314, 64274, 125561, 241569, 458715, 861242, 1601081, 2950693, 5396209, 9801012, 17692092, 31759800, 56727588, 100861716, 178585489, 314995915, 553650761, 969967510, 1694235803

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 (5,-7,-2,10,-2,-5,1,1).

Crossrefs

Cf. A046741, A002940, A002889, A055608, A062123-A062127.

Programs

Haskell

a002941 n = a002941_list !! (n-1)
a002941_list = 1 : 7 : 29 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a002941_list) a002941_list)
   (drop 2 $ zipWith (+) (tail a002940_list) a002940_list)
-- Reinhard Zumkeller, Jan 18 2014

Magma

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

Mathematica

CoefficientList[(1+x)^2/((1-x-x^2)^3*(1-x)^2) + O[x]^30, x] (* Jean-François Alcover, Jul 31 2018 *)
LinearRecurrence[{5,-7,-2,10,-2,-5,1,1},{1,7,29,94,263,667,1577,3538},30] (* Harvey P. Dale, Aug 29 2021 *)

PARI

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

Sage

((1+x)^2/((1-x-x^2)^3*(1-x)^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

Formula

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

a(n) = 2*a(n-1) - a(n-3) + A002940(n) + A002940(n-1).

Extensions

More terms from Henry Bottomley, Jun 02 2000