cp's OEIS Frontend

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

%I A052961 #64 Apr 04 2025 21:26:31
%S A052961 1,2,7,29,124,533,2293,9866,42451,182657,785932,3381689,14550649,
%T A052961 62608178,269388943,1159120181,4987434076,21459809837,92336746957,
%U A052961 397304305274,1709511285499,7355643511673,31649683701868,136181487974321,585958388766001,2521247479907042
%N A052961 Expansion of (1 - 3*x) / (1 - 5*x + 3*x^2).
%C A052961 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
%C A052961 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
%C A052961 A production matrix for the sequence is M =
%C A052961   1, 1, 0, 0, 0, 0, 0, ...
%C A052961   1, 0, 4, 0, 0, 0, 0, ...
%C A052961   1, 0, 0, 4, 0, 0, 0, ...
%C A052961   1, 0, 0, 0, 4, 0, 0, ...
%C A052961   1, 0, 0, 0, 0, 4, 0, ...
%C A052961   1, 0, 0, 0, 0, 0, 4, ...
%C A052961   ...
%C A052961 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
%C A052961 From _Gary W. Adamson_, Jul 29 2016: (Start)
%C A052961 The sequence is N=1 in an infinite set obtained from matrix powers of [(1,N); (1,4)], extracting the upper left terms.
%C A052961 The infinite set begins:
%C A052961 N=1  (A052961):  1,  2,  7,  29   124,  533,   2293, ...
%C A052961 N=2  (A052984):  1,  3, 13,  59,  269, 1227,   5597, ...
%C A052961 N=3  (A004253):  1,  4, 19,  91,  436, 2089,  10009, ...
%C A052961 N=4  (A000351):  1,  5, 25, 125,  625, 3125,  15625, ...
%C A052961 N=5  (A015449):  1,  6, 31, 161,  836, 4341,  22541, ...
%C A052961 N=6  (A124610):  1,  7, 37, 199, 1069, 5743,  30853, ...
%C A052961 N=7  (A111363):  1,  8, 43, 239, 1324, 7337,  40653, ...
%C A052961 N=8  (A123270):  1,  9, 49, 281, 1601, 9129,  52049, ...
%C A052961 N=9  (A188168):  1, 10, 55, 325, 1900, 11125, 65125, ...
%C A052961 N=10 (A092164):  1, 11, 61, 371, 2221, 13331, 79981, ...
%C A052961 ... (End)
%H A052961 Karl V. Keller, Jr., <a href="/A052961/b052961.txt">Table of n, a(n) for n = 0..1000</a>
%H A052961 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1032">Encyclopedia of Combinatorial Structures 1032</a>
%H A052961 Kai Liang, <a href="https://arxiv.org/abs/2503.17698">Solving tiling enumeration problems by tensor network contractions</a>, arXiv:2503.17698, March 2025.
%H A052961 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3).
%F A052961 G.f.: (1-3*x)/(1-5*x+3*x^2).
%F A052961 a(n) = 5*a(n-1) - 3*a(n-2), with a(0) = 1, a(1) = 2.
%F A052961 a(n) = Sum_{alpha=RootOf(1-5*z+3*z^2)} (-1 + 9*alpha)*alpha^(-1-n)/13.
%F A052961 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
%F A052961 a(n) = A116415(n) - 3*A116415(n-1). - _R. J. Mathar_, Feb 27 2019
%p A052961 spec:= [S,{S = Sequence(Union(Prod(Sequence(Union(Z,Z,Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size = n), n = 0..20);
%p A052961 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
%t A052961 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 *)
%o A052961 (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
%o A052961 (PARI) my(x='x+O('x^30)); Vec((1-3*x)/(1-5*x+3*x^2)) \\ _G. C. Greubel_, Oct 23 2019
%o A052961 (Sage)
%o A052961 def A052961_list(prec):
%o A052961     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052961     return P((1-3*x)/(1-5*x+3*x^2)).list()
%o A052961 A052961_list(30) # _G. C. Greubel_, Oct 23 2019
%o A052961 (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
%Y A052961 Column k=2 of A254414.
%Y A052961 Cf. A000351, A004253, A015449, A052984, A092164, A111363, A123270, A124610, A188168.
%K A052961 easy,nonn
%O A052961 0,2
%A A052961 encyclopedia(AT)pommard.inria.fr, Jan 25 2000