A135247 a(n) = 3*a(n-1) + 2*a(n-2) - 6*a(n-3).

1, 3, 11, 33, 103, 309, 935, 2805, 8431, 25293, 75911, 227733, 683263, 2049789, 6149495, 18448485, 55345711, 166037133, 498111911, 1494335733, 4483008223, 13449024669, 40347076055, 121041228165, 363123688591, 1089371065773, 3268113205511, 9804339616533

Offset: 0

Author

Paul Curtz, Feb 15 2008

Keywords

• nonn
• easy

Comments

This sequence interleaves A016133 and 3*A016133, see formulas. - Mathew Englander, Jan 08 2024

a(n) is the number of partitions of n into parts 1 (in three colors) and 2 (in two colors) where the order of colors matters. For example, the a(2)=11 such partitions (using parts 1, 1', 1'', 2, and 2') are 2, 2', 1+1, 1+1', 1+1'', 1'+1, 1'+1', 1'+1'', 1''+1, 1''+1', 1''+1''. For such partitions where the order of colors does not matter see A002624. - Joerg Arndt, Jan 18 2024

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,2,-6).

Crossrefs

Cf. A016133.

Programs

GAP

a:=[1,3,11];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-6*a[n-3]; od; a; # G. C. Greubel, Nov 20 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // G. C. Greubel, Nov 20 2019

Maple

seq(coeff(series(1/(1-3*x-2*x^2+6*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 20 2019

Mathematica

LinearRecurrence[{3,2,-6},{1,3,11},30] (* Harvey P. Dale, Jun 27 2015 *)

PARI

my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ G. C. Greubel, Nov 20 2019

Sage

def A135247_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
A135247_list(30) # G. C. Greubel, Nov 20 2019

Formula

G.f.: 1/((1-3*x)*(1-2*x^2)). - G. C. Greubel, Oct 04 2016

From Mathew Englander, Jan 08 2024: (Start)

a(n) = A010684(n) * A016133(floor(n/2)).

a(n) = 3*a(n-1) + A077957(n) for n >= 1.

a(n) = (A000244(n+2) - A164073(n+3))/7.

(End)

Extensions

More terms from Harvey P. Dale, Jun 27 2015

Dropped two leading terms = 0. - Joerg Arndt, Jan 18 2024