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.

A098608 a(n) = 100^n.

Original entry on oeis.org

1, 100, 10000, 1000000, 100000000, 10000000000, 1000000000000, 100000000000000, 10000000000000000, 1000000000000000000, 100000000000000000000, 10000000000000000000000, 1000000000000000000000000, 100000000000000000000000000, 10000000000000000000000000000, 1000000000000000000000000000000
Offset: 0

Views

Author

Henry Bottomley, Sep 17 2004

Keywords

Comments

For any base B, these are the numbers (B^2)^n written in base B. - Philippe Deléham, Jan 06 2008
Conjecture: a(n) are the only positive squares (in base 10) with digits in {0,1}. - Manfred Boergens, Feb 10 2025

References

  • Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A011557, A098609.

Programs

  • Mathematica
    LinearRecurrence[{100},{1},12] (* Ray Chandler, Aug 17 2015 *)
NestList[100#&,1,20] (* Harvey P. Dale, Dec 28 2018 *)

Formula

a(n) = 100*a(n-1) = A011557(2n) = A098609(n) + 1.
G.f.: 1/(1-100x).
E.g.f.: exp(100*x). - Stefano Spezia, Aug 05 2024

Extensions

a(12)-a(15) from Stefano Spezia, Aug 05 2024

A098610 a(n) = 10^n + (-1)^n.

Original entry on oeis.org

2, 9, 101, 999, 10001, 99999, 1000001, 9999999, 100000001, 999999999, 10000000001, 99999999999, 1000000000001, 9999999999999, 100000000000001, 999999999999999, 10000000000000001, 99999999999999999, 1000000000000000001, 9999999999999999999, 100000000000000000001
Offset: 0

Views

Author

Henry Bottomley, Sep 17 2004

Keywords

Links

Crossrefs

Cf. A015585, A094028, A098609, A098611.

Programs

  • Magma
    [10^n+(-1)^n: n in [0..20]]; // Vincenzo Librandi, Sep 23 2016
  • Mathematica
    Total/@Partition[Riffle[10^Range[0,20],{1,-1}],2] (* or *) Table[10^n+(-1)^n,{n,0,20}] (* Harvey P. Dale, Aug 20 2012 *)

Formula

a(n) = A098611(n) + 2*(-1)^n.
a(n) = A098609(n)/A098611(n).
a(n) = A098609(n)/(11*A015585(n)).
a(n) = 9*A094028(n+1)/A015585(n).
From Chai Wah Wu, Sep 22 2016: (Start)
a(n) = 9*a(n-1) + 10*a(n-2) for n > 1.
G.f.: (9*x - 2)/((x + 1)*(10*x - 1)). (End)
E.g.f.: exp(-x)*(exp(11*x) + 1). - Elmo R. Oliveira, Aug 17 2024

A098611 a(n) = 10^n - (-1)^n.

Original entry on oeis.org

0, 11, 99, 1001, 9999, 100001, 999999, 10000001, 99999999, 1000000001, 9999999999, 100000000001, 999999999999, 10000000000001, 99999999999999, 1000000000000001, 9999999999999999, 100000000000000001, 999999999999999999, 10000000000000000001
Offset: 0

Views

Author

Henry Bottomley, Sep 17 2004

Keywords

Links

Crossrefs

Cf. A015585, A098609, A098610.

Programs

  • Magma
    [10^n-(-1)^n : n in [0..30]]; // Wesley Ivan Hurt, Apr 05 2015
  • Maple
    A098611:=n->10^n-(-1)^n: seq(A098611(n), n=0..30); # Wesley Ivan Hurt, Apr 05 2015
  • Mathematica
    Table[(10^n - (-1)^n), {n, 0, 30}] (* or *)
CoefficientList[Series[11 x/(1 - 9 x - 10 x^2), {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 05 2015 *)
LinearRecurrence[{9, 10}, {0, 11}, 20] (* Vincenzo Librandi, Apr 06 2015 *)
  • PARI
    vector(20,n, 10^(n-1)+(-1)^n) \\ Derek Orr, Apr 05 2015

Formula

a(n) = A098610(n) - 2*(-1)^n = A098609(n)/A098610(n).
From Wesley Ivan Hurt, Apr 05 2015: (Start)
G.f.: 11*x/(1-9*x-10*x^2).
a(n) = 9*a(n-1) + 10*a(n-2). (End)
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(-x)*(exp(11*x) - 1).
a(n) = 11*A015585(n). (End)
Showing 1-3 of 3 results.

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