A027053 a(n) = T(n,n+2), T given by A027052.

1, 4, 9, 18, 35, 66, 123, 228, 421, 776, 1429, 2630, 4839, 8902, 16375, 30120, 55401, 101900, 187425, 344730, 634059, 1166218, 2145011, 3945292, 7256525, 13346832, 24548653, 45152014, 83047503, 152748174, 280947695, 516743376

Offset: 2

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Second differences of (A027026(n)-1)/2.

Pairwise sums of A089068.

a(n) is also the number of fixed polyominoes with n cells of height (or width) 2. - David Bevan, Sep 09 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1001
• Doron Zeilberger, The generating functions and series expansions for 2D lattice-animals of globally bounded width. [Local copies of generating functions and series expansions].
• Index entries for sequences related to polyominoes
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Crossrefs

2nd column of A308359.

Programs

GAP

a:=[1,4,9,18];; for n in [5..30] do a[n]:=2*a[n-1]-a[n-4]; od; a; # G. C. Greubel, Nov 05 2019

Magma

R:=PowerSeriesRing(Integers(), 32); Coefficients(R!( x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)) )); // G. C. Greubel, Nov 05 2019

Maple

seq(coeff(series(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)), x, n+1), x, n), n = 2 ..30); # G. C. Greubel, Nov 05 2019

Mathematica

LinearRecurrence[{2,0,0,-1}, {1,4,9,18}, 30] (* G. C. Greubel, Nov 05 2019 *)

PARI

my(x='x+O('x^32)); Vec(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3))) \\ G. C. Greubel, Nov 05 2019

Sage

def A027053_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3))).list()
a=A027053_list(32); a[2:] # G. C. Greubel, Nov 05 2019

Formula

G.f.: x^2*(1+x)^2/((1-x)*(1-x-x^2-x^3)).

a(n) = A089068(n-1) + A089068(n).

a(n) = a(n-1) + a(n-2) + a(n-3) + 4. - David Bevan, Sep 09 2009

a(n) = A001590(n+3) - 2. - David Bevan, Sep 09 2009

a(n+1) - a(n) = A000213(n+1). - R. J. Mathar, Aug 04 2013