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A027992 a(n) = 1*T(n,0) + 2*T(n,1) + ... + (2n+1)*T(n,2n), T given by A027926. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A099855 A003469 A189418 * A271389 A247168 A305032

Adjacent sequences:  A027989 A027990 A027991 * A027993 A027994 A027995

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified October 15 11:25 EDT 2018. Contains 316224 sequences. (Running on oeis4.)