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%I M3415 N1381 #66 Mar 01 2024 09:48:02
%S 1,4,11,26,56,114,223,424,789,1444,2608,4660,8253,14508,25343,44030,
%T 76136,131110,224955,384720,656041,1115784,1893216,3205416,5416441,
%U 9136084,15384563,25866914,43429784,72821274,121953943,204002680,340886973,569047468,949022608
%N Arrays of dumbbells.
%C 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
%C 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
%C 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
%C Partial Sums of A023610 - _John Molokach_, Jul 03 2013
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
%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).
%D D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H Reinhard Zumkeller, <a href="/A002940/b002940.txt">Table of n, a(n) for n = 1..1000</a>
%H Carlos Alirio Rico Acevedo and Ana Paula Chaves, <a href="https://arxiv.org/abs/1903.07490">Double-Recurrence Fibonacci Numbers and Generalizations</a>, arXiv:1903.07490 [math.NT], 2019.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See pp. 23-24.
%H R. C. Grimson, <a href="https://doi.org/10.1063/1.1666624">Exact formulas for 2 x n arrays of dumbbells</a>, J. Math. Phys., 15 (1974), 214-216.
%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 Y. Horibe, <a href="http://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
%H R. B. McQuistan and S. J. Lichtman, <a href="https://dx.doi.org/10.1063/1.1665098">Exact recursion relation for 2 x N arrays of dumbbells</a>, J. Math. Phys., 11 (1970), 3095-3099.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).
%F a(n) = 2*a(n-1) - a(n-3) + A000045(n+1).
%F G.f.: x*(1+x)/((1-x)*(1-x-x^2)^2).
%F a(n) = Sum_{k=0..n} ( Sum_{i=0..n} k*C(k, i-k) ). - _Paul Barry_, Feb 16 2005
%F E.g.f.: 2*exp(x) + exp(x/2)*((55*x - 50)*cosh(sqrt(5)*x/2) + sqrt(5)*(25*x - 22)*sinh(sqrt(5)*x/2))/25. - _Stefano Spezia_, Dec 03 2023
%t a[n_]:= a[n]= If[n<3, n^2, 2a[n-1] -a[n-3] +Fibonacci[n+1]]; Array[a, 32] (* _Jean-François Alcover_, Jul 31 2018 *)
%o (Haskell)
%o a002940 n = a002940_list !! (n-1)
%o a002940_list = 1 : 4 : 11 : zipWith (+)
%o (zipWith (-) (map (* 2) $ drop 2 a002940_list) a002940_list)
%o (drop 5 a000045_list)
%o -- _Reinhard Zumkeller_, Jan 18 2014
%o (PARI) my(x='x+O('x^35)); Vec((1+x)/((1-x)*(1-x-x^2)^2)) \\ _G. C. Greubel_, Jan 31 2019
%o (Magma) m:=35; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/((1-x)*(1-x-x^2)^2) )); // _G. C. Greubel_, Jan 31 2019
%o (Sage) ((1+x)/((1-x)*(1-x-x^2)^2)).series(x, 35).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 31 2019
%Y Cf. A002941, A002889, A055608, A062123, A062124, A062125, A062126, A062127, A046741.
%Y Cf. A000045, A001925, A023610, A054454, A006478, A067331, A178523.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Henry Bottomley_, Jun 02 2000
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