%I #60 Jul 29 2023 11:15:57
%S 0,2,3,8,15,30,56,104,189,340,605,1068,1872,3262,5655,9760,16779,
%T 28746,49096,83620,142065,240812,407353,687768,1159200,1950650,
%U 3277611,5499704,9216519,15426870,25793240,43080608,71884197,119835652
%N a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.
%C A Fibonacci-Lucas convolution.
%C The number of edges in the Lucas cube L_(n+1) [Klavzar]. - _R. J. Mathar_, Nov 05 2008
%C Sums of rows of the triangle in A108037. - _Reinhard Zumkeller_, Oct 07 2012
%C a(n-1) is the total binary weight of all bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - _Steven Finch_, May 26 2020
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 35.
%H Reinhard Zumkeller, <a href="/A099920/b099920.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Klavzar, <a href="http://dx.doi.org/10.1016/j.disc.2004.02.023">On median nature and enumerative properties of Fibonacci-like cubes</a>, Discr. Math. 299 (2005), 145-153.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1807.05256">A one-variable bracket polynomial for some Turk's head knots</a>, arXiv:1807.05256 [math.CO], 2018.
%H Steven Finch, <a href="https://arxiv.org/abs/2005.12185">Variance of longest run duration in a random bitstring</a>, arXiv:2005.12185 [math.CO], 2020.
%H Tamás Szakács, <a href="https://doi.org/10.1515/cm-2017-0011">Convolution of second order linear recursive sequences. II.</a> Commun. Math. 25, No. 2, 137-148 (2017), remark 2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasCubeGraph.html">Lucas Cube Graph</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1)
%F G.f.: x*(2-x)/(1-x-x^2)^2;
%F a(n) = Sum_{k=0..n} F(n-k)*(L(k-1) + 0^k).
%F a(n) = Sum_{k=0..n+1} F(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4); a(0)=0, a(1)=2, a(2)=3, a(3)=8. - _Harvey P. Dale_, Jan 18 2012
%F a(n) = a(n-1) + a(n-2) + A000032(n-1) (Lucas numbers). - _Bob Selcoe_, Aug 19 2015
%F a(n) = 2*A001629(n+1) - A001629(n). - _R. J. Mathar_, Feb 04 2022
%t Table[(n + 1) Fibonacci[n], {n, 0, 40}] (* _Harvey P. Dale_, Jan 18 2012 *)
%t LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 8}, 40] (* _Harvey P. Dale_, Jan 18 2012 *)
%t CoefficientList[Series[(2 - x) x/(-1 + x + x^2)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Jul 28 2023 *)
%o (Magma) [(n+1)*Fibonacci(n): n in [0..60]]; // _Vincenzo Librandi_, Apr 23 2011
%o (Haskell)
%o a099920 n = a099920_list !! n
%o a099920_list = zipWith (*) [1..] a000045_list
%o -- _Reinhard Zumkeller_, Oct 07 2012
%o (PARI) a(n)=(n+1)*fibonacci(n) \\ _Charles R Greathouse IV_, Jun 11 2015
%Y Equals A010049(n) + A001629(n+1).
%Y Cf. A000045, A000032, A045925, A023607.
%K nonn,easy
%O 0,2
%A _Paul Barry_ and _Ralf Stephan_, Oct 15 2004
%E Entry revised by _N. J. A. Sloane_, Jan 23 2006. The offset changed, so some of the formulas may now be slightly off.