A002941 Arrays of dumbbells. (Formerly M4396 N1852)

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

1,2

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

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).

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 *)

Prog

(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
(PARI) x='x+O('x^30); Vec((1+x)^2/((1-x-x^2)^3*(1-x)^2)) \\ Altug Alkan, Jul 31 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^2/((1-x-x^2)^3*(1-x)^2) )); // G. C. Greubel, Jan 31 2019
(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

Crossrefs

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

Sequence in context: A290901 A294843 A042609 * A193655 A369805 A227086

Adjacent sequences: A002938 A002939 A002940 * A002942 A002943 A002944

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Henry Bottomley, Jun 02 2000

Status

approved