A052961 Expansion of (1 - 3*x) / (1 - 5*x + 3*x^2).

1, 2, 7, 29, 124, 533, 2293, 9866, 42451, 182657, 785932, 3381689, 14550649, 62608178, 269388943, 1159120181, 4987434076, 21459809837, 92336746957, 397304305274, 1709511285499, 7355643511673, 31649683701868, 136181487974321, 585958388766001, 2521247479907042

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

a(n) is the number of tilings of a 2 X n rectangle using integer dimension tiles at least one of whose dimensions is 1, so allowable dimensions are 1 X 1, 1 X 2, 1 X 3, 1 X 4, ..., and 2 X 1. - David Callan, Aug 27 2014

a(n+1) counts closed walks on K_2 containing one loop on the index vertex and four loops on the other vertex. Equivalently the (1,1)entry of A^(n+1) where the adjacency matrix of digraph is A=(1,1;1,4). - _David Neil McGrath, Nov 05 2014

A production matrix for the sequence is M =

1, 1, 0, 0, 0, 0, 0, ...

1, 0, 4, 0, 0, 0, 0, ...

1, 0, 0, 4, 0, 0, 0, ...

1, 0, 0, 0, 4, 0, 0, ...

1, 0, 0, 0, 0, 4, 0, ...

1, 0, 0, 0, 0, 0, 4, ...

...

Take powers of M and extract the upper left term, getting the sequence starting (1, 1, 2, 7, 29, 124, ...). - Gary W. Adamson, Jul 22 2016

From Gary W. Adamson, Jul 29 2016: (Start)

The sequence is N=1 in an infinite set obtained from matrix powers of [(1,N); (1,4)], extracting the upper left terms.

The infinite set begins:

N=1 (A052961): 1, 2, 7, 29 124, 533, 2293, ...

N=2 (A052984): 1, 3, 13, 59, 269, 1227, 5597, ...

N=3 (A004253): 1, 4, 19, 91, 436, 2089, 10009, ...

N=4 (A000351): 1, 5, 25, 125, 625, 3125, 15625, ...

N=5 (A015449): 1, 6, 31, 161, 836, 4341, 22541, ...

N=6 (A124610): 1, 7, 37, 199, 1069, 5743, 30853, ...

N=7 (A111363): 1, 8, 43, 239, 1324, 7337, 40653, ...

N=8 (A123270): 1, 9, 49, 281, 1601, 9129, 52049, ...

N=9 (A188168): 1, 10, 55, 325, 1900, 11125, 65125, ...

N=10 (A092164): 1, 11, 61, 371, 2221, 13331, 79981, ...

... (End)

Links

• Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1032
• Kai Liang, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698, March 2025.
• Index entries for linear recurrences with constant coefficients, signature (5,-3).

Crossrefs

Column k=2 of A254414.

Cf. A000351, A004253, A015449, A052984, A092164, A111363, A123270, A124610, A188168.

Programs

GAP

a:=[1,2];; for n in [3..30] do a[n]:=5*a[n-1]-3*a[n-2]; od; a; # G. C. Greubel, Oct 23 2019

Magma

I:=[1,2]; [n le 2 select I[n] else 5*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 17 2014

Maple

spec:= [S,{S = Sequence(Union(Prod(Sequence(Union(Z,Z,Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size = n), n = 0..20);
seq(coeff(series((1-3*x)/(1-5*x+3*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 23 2019

Mathematica

CoefficientList[Series[(1-3x)/(1-5x+3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-3},{1,2},30] (* Harvey P. Dale, Nov 23 2013 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x)/(1-5*x+3*x^2)) \\ G. C. Greubel, Oct 23 2019

Sage

def A052961_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-5*x+3*x^2)).list()
A052961_list(30) # G. C. Greubel, Oct 23 2019

Formula

G.f.: (1-3*x)/(1-5*x+3*x^2).

a(n) = 5*a(n-1) - 3*a(n-2), with a(0) = 1, a(1) = 2.

a(n) = Sum_{alpha=RootOf(1-5*z+3*z^2)} (-1 + 9*alpha)*alpha^(-1-n)/13.

E.g.f.: (1 + sqrt(13) + (sqrt(13)-1) * exp(sqrt(13)*x)) / (2*sqrt(13) * exp(((sqrt(13)-5)*x)/2)). - Vaclav Kotesovec, Feb 16 2015

a(n) = A116415(n) - 3*A116415(n-1). - R. J. Mathar, Feb 27 2019