A027992 a(n) = 1*T(n,0) + 2*T(n,1) + ... + (2n+1)*T(n,2n), T given by A027926.

1, 6, 22, 66, 178, 450, 1090, 2562, 5890, 13314, 29698, 65538, 143362, 311298, 671746, 1441794, 3080194, 6553602, 13893634, 29360130, 61865986, 130023426, 272629762, 570425346, 1191182338, 2483027970, 5167382530, 10737418242

Offset

0,2

Comments

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

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.

Links

Table of n, a(n) for n=0..27.

Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, arXiv preprint arXiv:1110.5103 [math.CO], 2011.

Alejandro Erickson and Mark Schurch, Enumerating tatami mat arrangements of square grids, 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

K. Kimura, S. Higuchi, Monte Carlo estimation of the number of tatami tilings, arXiv:1509.05983 [cond-mat.stat-mech], 2015-2016, eq. (2).

Formula

a(n) = 2^n*(3n-1)+2 = A048496(n+1)-1 = A053565(n+1)+2. - Ralf Stephan, Jan 15 2004

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

Mathematica

M = {{1, 0, 0}, {1, 2, 0}, {1, 3, 2}};
a[n_] := MatrixPower[M, n].{1, 1, 1} // Last;
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 12 2018, from PARI *)

Prog

(PARI) vector(40, n, n--; ([1, 0, 0; 1, 2, 0; 1, 3, 2]^n*[1, 1, 1]~)[3]) \\ Michel Marcus, Aug 06 2015

Crossrefs

Cf. A027926, A066183.

Sequence in context: A003469 A364415 A189418 * A271389 A353703 A247168

Adjacent sequences: A027989 A027990 A027991 * A027993 A027994 A027995

Keyword

nonn

Author

Clark Kimberling

Status

approved