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.

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A215492 a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

Original entry on oeis.org

0, 3, 6, 63, 147, 1365, 3528, 29694, 83643, 648270, 1964361, 14199171, 45789471, 311933118, 1060973088, 6871121775, 24463966674, 151720368891, 561841152579, 3357375513429, 12860706786396, 74437773850062, 293576471108319, 1653218198356074, 6686170310225133
Offset: 0

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Author

Roman Witula, Aug 13 2012

Keywords

Comments

We have a(n)=B(n;3), where B(n;d), n=1,2,..., d \in C, denote one of the quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. Its conjugate sequences A(n;3) and C(n;3) are discussed in A121458 and A215484 respectively. Similarly as in A121458 we deduce that each of the following elements a(3*n), a(3*n+1), a(3*n+2) is divided by 3*7^n for every n=0,1,... .

Links

Crossrefs

Cf. A121458, A215484, A121449, A085810, A215404, A077998, A006054, A033304, A052975, A094789, A005021, A121442, A121458.

Programs

  • Magma
    I:=[0,3,6]; [n le 3 select I[n] else 21*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015
  • Mathematica
    LinearRecurrence[{0,21,7}, {0,3,6}, 50]
CoefficientList[Series[(3 x + 6 x^2)/(1 - 21 x^2 - 7 x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 18 2015 *)
  • PARI
    concat(0,Vec((3+6*x)/(1-21*x^2-7*x^3)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012

Formula

a(n) = (1/7)*((c(1)-c(4))*(1+3*c(1))^n + (c(2)-c(1))*(1+3*c(2))^n + (c(4)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper).
G.f.: (3*x+6*x^2)/(1-21*x^2-7*x^3).

A271945 Expansion of 4*x^2/(1 - x - 9*x^2 + x^3).

Original entry on oeis.org

0, 0, 4, 4, 40, 72, 428, 1036, 4816, 13712, 56020, 174612, 665080, 2180568, 7991676, 26951708, 96696224, 331269920, 1174584228, 4059317284, 14299305416, 49658576744, 174293008204, 606920893484, 2125899390576, 7413894423728, 25940068045428, 90539218468404
Offset: 0

Views

Author

Vincenzo Librandi, Apr 18 2016

Keywords

Links

Crossrefs

Cf. A121442, A121458, A215484, A215492, A271944.

Programs

  • Magma
    [n le 2 select 2*n*(n-1) else Self(n)+9*Self(n-1)-Self(n-2): n in [0..30]];
  • Mathematica
    CoefficientList[Series[4 x^2 /(1 - x - 9 x^2 + x^3), {x, 0, 30}], x]
LinearRecurrence[{1,9,-1},{0,0,4},30] (* Harvey P. Dale, Jul 18 2021 *)
  • PARI
    x='x+O('x^99); concat([0, 0], Vec(4*x^2/(1-x-9*x^2+x^3))) \\ Altug Alkan, Apr 18 2016
  • Sage
    gf = 4*x^2/(1 - x - 9*x^2 + x^3); taylor(gf, x, 0, 30).list() # Bruno Berselli, Apr 18 2016

Formula

G.f.: 4*x^2/(1 - x - 9*x^2 + x^3).
a(n) = a(n-1) + 9*a(n-2) - a(n-3).
Showing 1-2 of 2 results.

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