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.

A123005 Expansion of g.f. x^2/(1-2*x-49*x^2).

Original entry on oeis.org

0, 1, 2, 53, 204, 3005, 16006, 179257, 1142808, 11069209, 78136010, 698663261, 5225991012, 44686481813, 345446523214, 2880530655265, 22687940948016, 186521884004017, 1484752874460818, 12109078065118469, 96971046978817020
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Keywords

References

  • Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

Links

Crossrefs

Sequences of the form (m*i)^(n-1)*ChebyshevU(n-1, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), A123004 (m=5), this sequence (m=7), A123006 (m=11).

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) -49*Self(n-2): n in [1..31]]; // G. C. Greubel, Jul 12 2021
  • Mathematica
    CoefficientList[Series[x^2/(1-2x-49x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 12 2020 *)
  • Sage
    [(7*i)^(n-2)*chebyshev_U(n-2, -i/7) for n in [1..30]] # G. C. Greubel, Jul 12 2021

Formula

a(n) = 2*a(n-1) + 49*a(n-2).
a(n) = (7*i)^(n-2)*ChebyshevU(n-2, -i/7). - G. C. Greubel, Jul 12 2021

Extensions

Definition replaced by generating function - the Assoc. Eds. of the OEIS, Mar 27 2010

A123006 Expansion of x^2/(1 -2*x -121*x^2).

Original entry on oeis.org

0, 1, 2, 125, 492, 16109, 91750, 2132689, 15367128, 288789625, 2437001738, 39817548101, 374512306500, 5566947933221, 56449884952942, 786500469825625, 8403437018957232, 111973430886815089, 1240762741067455250
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Keywords

Links

Crossrefs

Sequences of the form (m*i)^(n-1)*ChebyshevU(n-1, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), A123004 (m=5), A123005 (m=7), this sequence (m=11).

Programs

  • Magma
    [n le 2 select n-1 else 2*Self(n-1) + 121*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 12 2021
  • Mathematica
    Rest@CoefficientList[Series[x^2/(1 -2*x -121*x^2), {x,0,30}], x]
  • Sage
    [(11*i)^(n-2)*chebyshev_U(n-2, -i/11) for n in [1..30]] # G. C. Greubel, Jul 12 2021

Formula

a(n) = 2*a(n-1) + 121*a(n-2).
a(n) = (11*i)^(n-2)*ChebyshevU(n-2, -i/11). - G. C. Greubel, Jul 12 2021

A123008 Expansion of x*(1 + 3*x)/(1 - 2*x - 25*x^2).

Original entry on oeis.org

1, 5, 35, 195, 1265, 7405, 46435, 277995, 1716865, 10383605, 63688835, 386967795, 2366156465, 14406507805, 87966927235, 536096549595, 3271366280065, 19945146300005, 121674449601635, 741977556703395, 4525816353447665
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Keywords

Links

Crossrefs

Cf. A002534, A123004.

Programs

  • Magma
    [n le 2 select 5^(n-1) else 2*Self(n-1) + 25*Self(n-2): n in [1..31]]; // G. C. Greubel, Jul 13 2021
  • Mathematica
    M:= {{0, 5}, {5, 2}}; v[1] = {1, 1}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 30}]
  • Sage
    [(5*i)^(n-2)*(3*chebyshev_U(n-2, -i/5) + 5*i*chebyshev_U(n-1, -i/5)) for n in (1..30)] # G. C. Greubel, Jul 13 2021

Formula

From Colin Barker, Oct 19 2012: (Start)
a(n) = 2*a(n-1) + 25*a(n-2) for n>2.
G.f.: x*(1+3*x)/(1-2*x-25*x^2). (End)
a(n) = (5*i)^(n-2)*(3*ChebyshevU(n-2, -i/5) + 5*i*ChebyshevU(n-1, -i/5)). - G. C. Greubel, Jul 13 2021

Extensions

Sequence edited by Joerg Arndt and Colin Barker, Oct 19 2012
Showing 1-3 of 3 results.

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