A092164 -id:A092164 - cp's OEIS frontend

A052961 Expansion of (1 - 3x) / (1 - 5x + 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

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

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)

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).

Column k=2 of A254414.

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

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

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

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

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 *)

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

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

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

A092165 Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (1,2) entry of M^n.

Original entry on oeis.org

2, 10, 62, 370, 2222, 13330, 79982, 479890, 2879342, 17276050, 103656302, 621937810, 3731626862, 22389761170, 134338567022, 806031402130, 4836188412782, 29017130476690, 174102782860142, 1044616697160850, 6267700182965102, 37606201097790610, 225637206586743662

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 01 2004

nonn

Index entries for linear recurrences with constant coefficients, signature (5,6).

Cf. A092164, A092166, A092167.

[(2*6^n - 2*(-1)^n)/7: n in [1..30]]; // Vincenzo Librandi, Jul 21 2015

Table[ MatrixPower[{{1, 2}, {5, 4}}, n][[1, 2]], {n, 20}] (* Robert G. Wilson v, Apr 22 2004 *)
LinearRecurrence[{5, 6}, {2, 10}, 25] (* Vincenzo Librandi, Jul 21 2015 *)

a(n) = (2*6^n - 2*(-1)^n)/7.
a(n) = A092164(n) -(-1)^n.
From R. J. Mathar, Apr 20 2009: (Start)
a(n) = 5*a(n-1) + 6*a(n-2) = 2*A015540(n).
G.f.: 2*x/((1+x)*(1-6*x)). (End)

Edited by Robert G. Wilson v, Apr 22 2004

More terms from Vincenzo Librandi, Jul 21 2015

A092166 Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (2,1) entry of M^n.

Original entry on oeis.org

5, 25, 155, 925, 5555, 33325, 199955, 1199725, 7198355, 43190125, 259140755, 1554844525, 9329067155, 55974402925, 335846417555, 2015078505325, 12090471031955, 72542826191725, 435256957150355, 2611541742902125

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 01 2004

Index entries for linear recurrences with constant coefficients, signature (5,6).

Cf. A092164, A092165, A092167.

Table[ MatrixPower[{{1, 2}, {5, 4}}, n][[2, 1]], {n, 20}] (* Robert G. Wilson v, Apr 22 2004 *)

a(n) = (5*6^n - 5*(-1)^n)/7.
a(n) = A092167(n) +(-1)^n.
From Colin Barker, Nov 08 2012: (Start)
a(n) = 5*a(n-1) + 6*a(n-2).
G.f.: -5*x/((x+1)*(6*x-1)). (End)

Edited by Robert G. Wilson v, Apr 22 2004

A092167 Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (2,2) entry of M^n.

Original entry on oeis.org

4, 26, 154, 926, 5554, 33326, 199954, 1199726, 7198354, 43190126, 259140754, 1554844526, 9329067154, 55974402926, 335846417554, 2015078505326, 12090471031954, 72542826191726, 435256957150354, 2611541742902126

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 01 2004

nonn

Index entries for linear recurrences with constant coefficients, signature (5, 6).

Cf. A092164, A092165, A092166.

Table[ MatrixPower[{{1, 2}, {5, 4}}, n][[2, 2]], {n, 20}] (* Robert G. Wilson v, Apr 22 2004 *)

a(n) = (5*6^n + 2*(-1)^n)/7.
A092166 -(-1)^n.
From R. J. Mathar, Apr 20 2009: (Start)
a(n) = 5*a(n-1) + 6*a(n-2).
G.f.: 2*x*(2+3*x)/((1+x)*(1-6*x)). (End)

Edited by Robert G. Wilson v, Apr 22 2004

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A092165 Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (1,2) entry of M^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A092166 Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (2,1) entry of M^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A092167 Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (2,2) entry of M^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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