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 1-4 of 4 results.

A122100 a(n) = 3*a(n-1) - a(n-3) for n>2, with a(0)=1, a(1)=-1, a(2)=0.

Original entry on oeis.org

1, -1, 0, -1, -2, -6, -17, -49, -141, -406, -1169, -3366, -9692, -27907, -80355, -231373, -666212, -1918281, -5523470, -15904198, -45794313, -131859469, -379674209, -1093228314, -3147825473, -9063802210, -26098178316, -75146709475, -216376326215, -623030800329, -1793945691512
Offset: 0

Views

Author

Philippe Deléham, Oct 18 2006

Keywords

Links

Crossrefs

Cf. A052536, A122099.

Programs

  • GAP
    a:=[1,-1,0];; for n in [4..40] do a[n]:=3*a[n-1]-a[n-3]; od; a; # G. C. Greubel, Oct 02 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-4*x+3*x^2)/(1-3*x+x^3) )); // G. C. Greubel, Oct 02 2019
  • Maple
    seq(coeff(series((1-4*x+3*x^2)/(1-3*x+x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 02 2019
  • Mathematica
    LinearRecurrence[{3,0,-1},{1,-1,0},40] (* Harvey P. Dale, Nov 14 2014 *)
  • PARI
    Vec((1-4*x+3*x^2)/(1-3*x+x^3)+O(x^40)) \\ Charles R Greathouse IV, Jan 17 2012
  • Sage
    def A122100_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x+3*x^2)/(1-3*x+x^3)).list()
A122100_list(40) # G. C. Greubel, Oct 02 2019

Formula

G.f.: (1-4*x+3*x^2)/(1-3*x+x^3).

A123942 The (1,4)-entry in the 4 X 4 matrix M^n, where M={{3, 2, 1, 1}, {2, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}} (n>=0).

Original entry on oeis.org

0, 1, 3, 15, 71, 340, 1626, 7778, 37205, 177966, 851280, 4072001, 19477953, 93170570, 445670811, 2131815570, 10197297001, 48777608903, 233322137235, 1116069871981, 5338593130960, 25536552265626, 122151189577128, 584296304368075, 2794914830384226
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

Keywords

References

  • Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26

Links

Crossrefs

Cf. A122099, A122100.

Programs

  • GAP
    a:=[0,1,3,15];; for n in [5..30] do a[n]:=4*a[n-1]+4*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Aug 05 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4) )); // G. C. Greubel, Aug 05 2019
  • Maple
    with(linalg): M[1]:=matrix(4,4,[3,2,1,1,2,1,1,0,1,1,0,0,1,0,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0, seq(M[n][1,4], n=1..30);
a[0]:=0: a[1]:=1: a[2]:=3: a[3]:=15: for n from 4 to 30 do a[n]:=4*a[n-1] +4*a[n-2]-a[n-3]-a[n-4] od: seq(a[n], n=0..30);
  • Mathematica
    M = {{3,2,1,1}, {2,1,1,0}, {1,1,0,0}, {1,0,0,0}}; v[1] = {0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
LinearRecurrence[{4,4,-1,-1}, {0,1,3,15}, 30] (* G. C. Greubel, Aug 05 2019 *)
  • PARI
    concat([0], Vec(x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4) + O(x^30))) \\ Colin Barker, Oct 18 2013
  • Sage
    (x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

Formula

a(n) = 4*a(n-1) + 4*a(n-2) - a(n-3) - a(n-4) for n>=4 (follows from the minimal polynomial of the matrix M).
G.f.: x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4). - Colin Barker, Oct 18 2013

Extensions

Edited by N. J. A. Sloane, Dec 04 2006
More terms from Colin Barker, Oct 18 2013

A123941 The (1,2)-entry in the 3 X 3 matrix M^n, where M = {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}.

Original entry on oeis.org

0, 1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977, 8462447, 24366645, 70160958, 202020427, 581694636, 1674922950, 4822748423, 13886550633, 39984728949, 115131438424, 331507764639, 954538564968, 2748484256480
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

Keywords

Comments

Essentially the same as A076264. - Tom Edgar, May 12 2015

References

  • Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26

Links

Crossrefs

Cf. A018919, A122099, A122100.

Programs

  • GAP
    a:=[0,1,3];; for n in [4..30] do a[n]:=3*a[n-1]-a[n-3]; od; a; # Muniru A Asiru, Oct 28 2018
  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x/(1-3*x+x^3) )); // G. C. Greubel, Aug 05 2019
  • Maple
    with(linalg): M[1]:=matrix(3,3,[2,1,1,1,1,0,1,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,2], n=1..30);
a[0]:=0: a[1]:=1: a[2]:=3: for n from 3 to 30 do a[n]:=3*a[n-1]-a[n-3] od: seq(a[n], n=0..30);
  • Mathematica
    M = {{2,1,1}, {1,1,0}, {1,0,0}}; v[1] = {0,0,1}; v[n_]:= v[n] =M.v[n-1];Table[v[n][[2]], {n, 30}]
LinearRecurrence[{3,0,-1}, {0,1,3}, 30] (* G. C. Greubel, Aug 05 2019 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+x^3))) \\ G. C. Greubel, Aug 05 2019
  • Sage
    (x/(1-3*x+x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

Formula

a(n) = 3*a(n-1) - a(n-3), a(0)=0, a(1)=1, a(2)=3 (follows from the minimal polynomial x^3-3x^2+1 of the matrix M).
a(n) = A076264(n-1). - R. J. Mathar, Jun 18 2008
G.f.: x/(1 - 3*x + x^3). - Arkadiusz Wesolowski, Oct 29 2012
a(n) = A018919(n-2) for n >= 2. - Georg Fischer, Oct 28 2018

Extensions

Edited by N. J. A. Sloane, Nov 07 2006

A123943 The (1,5)-entry in the 5 X 5 matrix M^n, where M={{5, 3, 2, 1, 1}, {3, 2, 1, 1, 0}, {2, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}}.

Original entry on oeis.org

0, 1, 5, 40, 315, 2490, 19681, 155563, 1229604, 9719061, 76821600, 607214857, 4799560053, 37936780428, 299860673343, 2370164848026, 18734305316497, 148080078051971, 1170457572108040, 9251554605638681, 73126326541645648, 578006601205833441
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

Keywords

References

Links

Crossrefs

Cf. A122099, A122100.

Programs

  • GAP
    a:=[0,1,5,40,315];; for n in [6..30] do a[n]:=8*a[n-1]-6*a[n-3] +a[n-5]; od; a; # G. C. Greubel, Aug 05 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) )); // G. C. Greubel, Aug 05 2019
  • Maple
    with(linalg): M[1]:=matrix(5,5,[5,3,2,1,1,3,2,1,1,0,2,1,1,0,0,1,1,0,0,0,1,0,0,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,5],n=1..30);
a[0]:=0: a[1]:=1: a[2]:=5: a[3]:=40: a[4]:=315: for n from 5 to 30 do a[n]:=8*a[n-1]-6*a[n-3]+a[n-5] od: seq(a[n],n=0..30);
# third Maple program:
a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>,
         <1|0|-6|0|8>>^n. <<0, 1, 5, 40, 315>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 05 2019
  • Mathematica
    M = {{5,3,2,1,1}, {3,2,1,1,0}, {2,1,1,0,0}, {1,1,0,0,0}, {1,0,0,0,0}}; v[1] = {0,0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
LinearRecurrence[{8,0,-6,0,1}, {0,1,5,40,315}, 30] (* G. C. Greubel, Aug 05 2019 *)
  • PARI
    concat(0, Vec(x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) + O(x^30))) \\ Colin Barker, Mar 03 2017
  • Sage
    (x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

Formula

a(n) = 8*a(n-1) - 6*a(n-3) + a(n-5) for n>=5 (follows from the minimal polynomial of the matrix M).
G.f.: x*(1 - 3*x + x^3) / (1 - 8*x + 6*x^3 - x^5). - Colin Barker, Mar 03 2017

Extensions

Edited by N. J. A. Sloane, Dec 04 2006
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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