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A002940 Arrays of dumbbells.
(Formerly M3415 N1381)
16
1, 4, 11, 26, 56, 114, 223, 424, 789, 1444, 2608, 4660, 8253, 14508, 25343, 44030, 76136, 131110, 224955, 384720, 656041, 1115784, 1893216, 3205416, 5416441, 9136084, 15384563, 25866914, 43429784, 72821274, 121953943, 204002680 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Whitney transform of n. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - Paul Barry, Feb 16 2005

a(n-1) is the permanent of the n X n 0-1 matrix with 1 in (i,j) position iff (i=1 and j<n) or 0<=i-j<=2 or (j=n and i>1). For example, with n=5, a(4) = per([[1, 1, 1, 1, 0], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1]]) = 26. - David Callan, Jun 07 2006

a(n) is the internal path length of the Fibonacci tree of order n+2. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The internal path length of a tree is the sum of the levels of all of its internal (i.e. non-leaf) nodes. [Emeric Deutsch, Jun 15 2010]

Partial Sums of A023610 - John Molokach, Jul 03 2013

REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).

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).

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. [Emeric Deutsch, Jun 15 2010]

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 (1974), 214-216.

R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)

Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. [Emeric Deutsch, Jun 15 2010]

R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1)

FORMULA

a(n) = 2*a(n-1) - a(n-3) + A000045(n+1).

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

a(n) = sum{k=0..n, sum{i=0..n, C(k, i-k)}*k}. - Paul Barry, Feb 16 2005

MATHEMATICA

a[n_] := a[n] = If[n<3, n^2, 2 a[n-1] - a[n-3] + Fibonacci[n+1]];

Array[a, 32] (* Jean-Fran├žois Alcover, Jul 31 2018 *)

PROG

(Haskell)

a002940 n = a002940_list !! (n-1)

a002940_list = 1 : 4 : 11 : zipWith (+)

   (zipWith (-) (map (* 2) $ drop 2 a002940_list) a002940_list)

   (drop 5 a000045_list)

-- Reinhard Zumkeller, Jan 18 2014

CROSSREFS

Cf. A002941, A002889, A055608, A062123-A062127, A046741.

Cf. A001925, A054454, A006478.

Cf. A067331, A178523. - Emeric Deutsch, Jun 15 2010

Sequence in context: A192961 A290989 A027660 * A030196 A248425 A130103

Adjacent sequences:  A002937 A002938 A002939 * A002941 A002942 A002943

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Henry Bottomley, Jun 02 2000

STATUS

approved

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Last modified October 15 05:56 EDT 2018. Contains 316202 sequences. (Running on oeis4.)