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%I #55 Apr 17 2023 02:06:35
%S 1,3,8,18,39,81,166,336,677,1359,2724,5454,10915,21837,43682,87372,
%T 174753,349515,699040,1398090,2796191,5592393,11184798,22369608,
%U 44739229,89478471,178956956,357913926,715827867,1431655749,2863311514,5726623044,11453246105,22906492227,45812984472,91625968962
%N Expansion of 1/((1-x)*(1-2*x)*(1-x^2)).
%H Vincenzo Librandi, <a href="/A011377/b011377.txt">Table of n, a(n) for n = 0..1000</a>
%H Kenneth Edwards and Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Allen/edwards2.html">New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile</a>, J. Int. Seq. 24 (2021) Article 21.3.8.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,2).
%F From _Paul Barry_, Jul 29 2004: (Start)
%F a(n) = Sum_{k=0..n+2} floor((n-k+2)/2) * 2^k;
%F a(n) = Sum_{k=0..n+2} floor(k/2) * 2^(n-k+2). (End)
%F a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n-k+2, k+2)*2^k. - _Paul Barry_, Oct 25 2004
%F a(n) = floor((2^(n+4) - 3*n - 6)/6). - _David W. Wilson_, Feb 26 2006
%F a(n) = (2^(n+5) - 6*n - 21 + (-1)^n)/12. - _Hieronymus Fischer_, Dec 02 2006
%F Row sums of triangle A135086. - _Gary W. Adamson_, Nov 18 2007
%F a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4). - _Paul Curtz_, Jul 29 2008
%F G.f.: Q(0)/(3*x*(1-x)^2), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - _Sergei N. Gladkovskii_, May 21 2013
%F From _Michael A. Allen_, Jan 11 2022: (Start)
%F a(n) = J(n+3) - ceiling((n+3)/2), where Jacobsthal number J(n) = A001045(n).
%F a(n) = Sum_{j=1..n+1} j*J(n+2-j). (End)
%t Table[(2^(n+5) -6*n-21+(-1)^n)/12, {n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2011, modified by _G. C. Greubel_, Jun 02 2019 *)
%t CoefficientList[Series[1/((1-x)(1-2x)(1-x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-3,2},{1,3,8,18},30] (* _Harvey P. Dale_, Apr 17 2017 *)
%o (Magma) [Floor((2^(n+4)-3*n-6)/6): n in [0..30]]; // _Vincenzo Librandi_, Aug 14 2011
%o (PARI) my(x='x+O('x^30)); Vec(1/((1-x)*(1-2*x)*(1-x^2))) \\ _G. C. Greubel_, Sep 26 2017
%o (Sage) [(2^(n+5) -6*n-21+(-1)^n)/12 for n in (0..30)] # _G. C. Greubel_, Jun 02 2019
%o (GAP) List([0..30], n-> (2^(n+5) -6*n -21 +(-1)^n)/12) # _G. C. Greubel_, Jun 02 2019
%Y Partial sums of A000975.
%Y Second partial sums of A001045.
%Y Cf. A135086.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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