A053565 -id:A053565 - cp's OEIS frontend

A027992 a(n) = 1T(n,0) + 2T(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

Clark Kimberling

nonn

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.

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).

Cf. A027926, A066183.

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 *)

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

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

A048496 a(n) = 2^(n-1)(3n-4) + 3.

Original entry on oeis.org

1, 2, 7, 23, 67, 179, 451, 1091, 2563, 5891, 13315, 29699, 65539, 143363, 311299, 671747, 1441795, 3080195, 6553603, 13893635, 29360131, 61865987, 130023427, 272629763, 570425347, 1191182339, 2483027971, 5167382531

Offset: 0

Clark Kimberling

a(n) = T(2, n), array T given by A048494.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5, -8, 4).

n-th difference of a(n), a(n-1), ..., a(0) is (1, 4, 7, 10, ...).

[2^(n-1)*(3*n-4)+3 : n in [0..30]]; // Vincenzo Librandi, Sep 26 2011

a(n) = A027992(n-1) + 1 = A053565(n) + 3.
From R. J. Mathar, Oct 31 2008: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).
G.f.: (1 - 3*x + 5*x^2)/((1-x)(1-2*x)^2). (End)

Formula from Ralf Stephan, Jan 15 2004

cp's OEIS Frontend

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

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A048496 a(n) = 2^(n-1)(3n-4) + 3.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A048496 a(n) = 2^(n-1)*(3*n-4) + 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

Extensions

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

A048496 a(n) = 2^(n-1)(3n-4) + 3.