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A002941 Arrays of dumbbells.
(Formerly M4396 N1852)
12
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 (list; graph; refs; listen; history; text; internal format)
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
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
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
Sequence in context: A290901 A294843 A042609 * A193655 A369805 A227086
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Henry Bottomley, Jun 02 2000
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)