Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M4715 N2016 #35 Sep 08 2022 08:44:31
%S 1,10,56,234,815,2504,7018,18336,45328,107160,244198,539656,1161987,
%T 2446906,5054440,10266850,20549117,40595568,79271188,153190480,
%U 293278496,556737696,1048772300,1961855408,3646420325,6737649754
%N Arrays of dumbbells.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
%D R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
%D R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A002889/b002889.txt">Table of n, a(n) for n = 1..1000</a>
%H R. C. Grimson, <a href="/A002889/a002889.pdf">Exact formulas for 2 x n arrays of dumbbells</a>, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (7,-17,11,19,-29,-3,21,-3,-7,1,1).
%F a(n) = 2*a(n-1) - a(n-3) + A002941(n) + A002941(n-1).
%F G.f.: (1+x)^3/((1-x)^3*(1-x-x^2)^4).
%t CoefficientList[(1+x)^3/((1-x)^3*(1-x-x^2)^4) + O[x]^30, x] (* _Jean-François Alcover_, Jul 31 2018 *)
%t 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 *)
%o (Haskell)
%o a002889 n = a002889_list !! (n-1)
%o a002889_list = 1 : 10 : 56 : zipWith (+)
%o (zipWith (-) (map (* 2) $ drop 2 a002889_list) a002889_list)
%o (drop 2 $ zipWith (+) (tail a002941_list) a002941_list)
%o -- _Reinhard Zumkeller_, Jan 18 2014
%o (PARI) x='x+O('x^30); Vec((1+x)^3/((1-x)^3*(1-x-x^2)^4)) \\ _Altug Alkan_, Jul 31 2018
%o (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
%o (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
%Y Cf. A046741, A002940, A002941.
%Y Cf. A055608, A062123-A062127.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Henry Bottomley_, Jun 02 2000