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%I #49 Jun 27 2022 21:19:03
%S 1,6,22,66,178,450,1090,2562,5890,13314,29698,65538,143362,311298,
%T 671746,1441794,3080194,6553602,13893634,29360130,61865986,130023426,
%U 272629762,570425346,1191182338,2483027970,5167382530,10737418242
%N a(n) = 1*T(n,0) + 2*T(n,1) + ... + (2n+1)*T(n,2n), T given by A027926.
%C Also total sum of squares of parts in all compositions of n (offset 1). Total sum of cubes of parts in all compositions of n is (13*n-36)*2^(n-1)+6*n+18 with g.f. x*(1+4x+x^2)/((2x-1)(1-x))^2, A271638; total sum of fourth powers of parts in all compositions of n is (75*n-316)*2^(n-1)+12*n^2+72*n+158 with g.f. x*(1+x)*(x^2+10*x+1)/((2*x-1)^2*(1-x)^3); total sum of fifth powers of parts in all compositions of n is (541*n-3060)*2^(n-1)+20*n^3+180*n^2+790*n+1530. - _Vladeta Jovovic_, Mar 18 2005
%C Let M = the 3 X 3 matrix [(1,0,0),(1,2,0),(1,3,2)] and column vector V = [1,1,1]. a(n) is the lower term in the product M^n * V.
%H Alejandro Erickson and Mark Schurch, <a href="http://arxiv.org/abs/1110.5103">Monomer-dimer tatami tilings of square regions</a>, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
%H Alejandro Erickson and Mark Schurch, <a href="http://dx.doi.org/10.1007/978-3-642-25011-8_18">Enumerating tatami mat arrangements of square grids</a>, in 22nd International Workshop on Combinatorial Algorithms, University of Victoria, June 20-22, volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, pp. 223-235
%H K. Kimura, S. Higuchi, <a href="http://arxiv.org/abs/1509.05983">Monte Carlo estimation of the number of tatami tilings</a>, arXiv:1509.05983 [cond-mat.stat-mech], 2015-2016, eq. (2).
%F a(n) = 2^n*(3n-1)+2 = A048496(n+1)-1 = A053565(n+1)+2. - _Ralf Stephan_, Jan 15 2004
%F a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). G.f.: (1+x)/((1-x)*(1-2*x)^2). - _Colin Barker_, Apr 04 2012
%t M = {{1, 0, 0}, {1, 2, 0}, {1, 3, 2}};
%t a[n_] := MatrixPower[M, n].{1, 1, 1} // Last;
%t Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Aug 12 2018, from PARI *)
%o (PARI) vector(40, n, n--; ([1,0,0;1,2,0;1,3,2]^n*[1,1,1]~)[3]) \\ _Michel Marcus_, Aug 06 2015
%Y Cf. A027926, A066183.
%K nonn
%O 0,2
%A _Clark Kimberling_
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