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A052961 Expansion of (1 - 3*x) / (1 - 5*x + 3*x^2). 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

Index entries for linear recurrences with constant coefficients, signature (5,-3).

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

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 *)

PROG

(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

(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.<x> = PowerSeriesRing(ZZ, prec)

    return P((1-3*x)/(1-5*x+3*x^2)).list()

A052961_list(30) # G. C. Greubel, Oct 23 2019

(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

CROSSREFS

Column k=2 of A254414.

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

Sequence in context: A263367 A120757 A134169 * A150662 A278391 A126568

Adjacent sequences:  A052958 A052959 A052960 * A052962 A052963 A052964

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified April 6 05:38 EDT 2020. Contains 333267 sequences. (Running on oeis4.)