cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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

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

A100638 Successive powers of the matrix A=[1,2;3,4] written by rows in groups of 4.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 15, 22, 37, 54, 81, 118, 199, 290, 435, 634, 1069, 1558, 2337, 3406, 5743, 8370, 12555, 18298, 30853, 44966, 67449, 98302, 165751, 241570, 362355, 528106, 890461, 1297782, 1946673, 2837134, 4783807, 6972050, 10458075
Offset: 1

Views

Author

Simone Severini, Dec 04 2004

Keywords

Comments

Consider the matrix A = [1, 2; 3, 4]. Then the sequence gives a(1) = A_{1,1} = A_11, a(2) = A_12, a(3) = A_21, a(4) = A_22, a(5)=(A^2)_11, a(6)=(A^2)_12, a(7)=(A^2)_21, a(8)=(A^2)_22, a(9)=(A^3)_11, a(10)=(A^3)_12, ...

Links

Programs

  • Maple
    a:= proc(n) local r, m; (<<1|2>, <3|4>>^iquo(n+3, 4, 'r'))[iquo(r+2, 2, 'm'), m+1] end: seq(a(n), n=1..50); # Alois P. Heinz, Dec 01 2008
  • Mathematica
    LinearRecurrence[{0, 0, 0, 5, 0, 0, 0, 2}, {1, 2, 3, 4, 7, 10, 15, 22}, 50] (* Jean-François Alcover, May 18 2018, after R. J. Mathar *)
  • PARI
    A100638(n)=([1,2;3,4]^((n-1)\4+1))[(n-1)%4\2+1,2-n%2] /* M. F. Hasler, Dec 01 2008 */

Formula

a(4n-3) = A124610(n), a(4n-2) = 2 A015535(n), a(4n-1) = 3 A015535(n), a(4n) = a(4n-3) + a(4n-1). - M. F. Hasler, Dec 01 2008
a(n) = 5*a(n-4)+2*a(n-8). a(4n+1)=A124610(n+1), n>=0. G.f.: x*(1+2x+3x^2+4x^3+2x^4+2x^7) / (1-5x^4-2x^8). - R. J. Mathar, Dec 04 2008

Extensions

Edited by Benoit Jubin, M. F. Hasler and N. J. A. Sloane, Dec 01 2008

A327765 a(n) is the trace of the n-th power of the 2 X 2 matrix [1 2; 3 4].

Original entry on oeis.org

2, 5, 29, 155, 833, 4475, 24041, 129155, 693857, 3727595, 20025689, 107583635, 577969553, 3105015035, 16681014281, 89615101475, 481437535937, 2586417882635, 13894964485049, 74647658190515, 401028219922673, 2154436415994395, 11574238519817321, 62180065431075395
Offset: 0

Views

Author

Adolf Cusmariu, Sep 24 2019

Keywords

Links

Crossrefs

Cf. A100638, A124610, A233020.

Programs

  • Maple
    a:= n-> (<<0|1>, <2|5>>^n. <<2, 5>>)[1, 1]:
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 07 2019
  • Mathematica
    CoefficientList[Series[(2 - 5 x)/(1 - 5 x - 2 x^2), {x, 0, 22}], x] (* Michael De Vlieger, Sep 27 2019 *)
LinearRecurrence[{5,2},{2,5},30] (* Harvey P. Dale, Jun 25 2020 *)
  • PARI
    a(n)={trace([1,2;3,4]^n)} \\ Andrew Howroyd, Sep 24 2019
  • PARI
    Vec((2 - 5*x) / (1 - 5*x - 2*x^2) + O(x^25)) \\ Colin Barker, Sep 27 2019

Formula

a(n) = trace(M^n) where M is [1, 2; 3, 4].
From Colin Barker, Sep 27 2019: (Start)
G.f.: (2 - 5*x) / (1 - 5*x - 2*x^2).
a(n) = 5*a(n-1) + 2*a(n-2) for n > 1.
a(n) = ((5-sqrt(33))/2)^n + ((5+sqrt(33))/2)^n.
(End)
Showing 1-3 of 3 results.

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