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.

A010903 Pisot sequence E(3,13): a(n) = floor(a(n-1)^2/a(n-2) + 1/2).

Original entry on oeis.org

3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134
Offset: 0

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Author

Simon Plouffe

Keywords

Comments

According to Boyd (Acta Arithm. 32 (1977) p 89), quoting Pisot, every E(3,.) sequence satisfies a linear recurrence of at most order 3. Here this is easily derived from the first terms of the sequence. - R. J. Mathar, May 26 2008
A010920 coincides with this sequence for at least the first 32600 terms and probably more. - R. J. Mathar, May 26 2008
For n >= 1, a(n-1) is the number of generalized compositions of n when there are i+2 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Links

Crossrefs

Cf. A010920.

Programs

  • Mathematica
    LinearRecurrence[{5, -3}, {3, 13}, 24] (* Jean-François Alcover, Oct 22 2019 *)
nxt[{a_,b_}]:={b,Floor[b^2/a+1/2]}; NestList[nxt,{3,13},30][[;;,1]] (* Harvey P. Dale, Mar 29 2025 *)
  • PARI
    Vec((3-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Jul 27 2016

Formula

a(n) = 5*a(n-1) - 3*a(n-2) = 3*A116415(n) - 2*A116415(n-1). - R. J. Mathar, May 26 2008
O.g.f.: (3-2*x)/(1-5*x+3*x^2). - R. J. Mathar, May 26 2008
a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-11+3*sqrt(13)) + (5+sqrt(13))^n*(11+3*sqrt(13))))/sqrt(13). - Colin Barker, Nov 26 2016

A095934 Expansion of (1-x)^2/(1-5*x+3*x^2).

Original entry on oeis.org

1, 3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134
Offset: 0

Views

Author

N. J. A. Sloane, Jul 13 2004

Keywords

Comments

a(n) is the number of generalized compositions of n when there are i+2 different types of i, (i=1,2,...). [Milan Janjic, Sep 24 2010]

Links

Crossrefs

Cf. A003480, A010903, A010920; equals A095939 + A095940.

Programs

  • Mathematica
    CoefficientList[Series[(1-x)^2/(1-5x+3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-3},{1,3,13},30] (* Harvey P. Dale, Jun 21 2021 *)
  • PARI
    a(n)=polcoeff((1-x)^2/(1-5*x+3*x^2)+x*O(x^n),n)

Formula

a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 1, a(1) = 3, a(2) = 13.

A278681 Pisot sequence T(3,16).

Original entry on oeis.org

3, 16, 85, 451, 2392, 12686, 67280, 356818, 1892376, 10036172, 53226604, 282286052, 1497097488, 7939821584, 42108658448, 223322287224, 1184384537744, 6281355751296, 33313023614352, 176674843181968, 936990907061504, 4969309405367264, 26354616443092800, 139771093164846816, 741272730213321216, 3931322622695991104
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2016

Keywords

Links

Crossrefs

Cf. A008776 for definitions of Pisot sequences.
Cf. A010920, A018920.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 3, a[1] == 16, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 25}]

Formula

a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 3, a(1) = 16.
Conjectures: (Start)
G.f.: (3 - 2*x + x^2 - x^3)/(1 - 6*x + 4*x^2 - 2*x^3 + 2*x^4).
a(n) = 6*a(n-1) - 4*a(n-2) + 2*a(n-3) - 2*a(n-4). (End)
Showing 1-3 of 3 results.

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

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