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.

A093136 Expansion of (1 - 8*x)/(1 - 10*x).

Original entry on oeis.org

1, 2, 20, 200, 2000, 20000, 200000, 2000000, 20000000, 200000000, 2000000000, 20000000000, 200000000000, 2000000000000, 20000000000000, 200000000000000, 2000000000000000, 20000000000000000, 200000000000000000, 2000000000000000000, 20000000000000000000
Offset: 0

Views

Author

Paul Barry, Mar 24 2004

Keywords

Comments

A convex combination of 10^n and 0^n.
Inverse binomial transform of A083294. - Stefano Spezia, Jul 07 2021

Links

Crossrefs

Partial sums are A093135.
Cf. A083294, A132026.

Programs

  • Mathematica
    CoefficientList[Series[(1-8x)/(1-10x),{x,0,30}],x] (* or *) LinearRecurrence[{10},{1,2},30] (* Harvey P. Dale, Oct 02 2022 *)
  • PARI
    Vec((1-8*x)/(1-10*x) + O(x^20)) \\ Felix Fröhlich, Jul 07 2021

Formula

a(n) = (2*10^n + 8*0^n)/10.
a(n) = 2*10^(n-1), n > 0. - Vincenzo Librandi, Aug 02 2010
E.g.f.: (8 + 2*exp(10*x))/10. - Stefano Spezia, Jul 05 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 14/9.
Sum_{n>=0} (-1)^n/a(n) = 6/11.
Product_{n>=1} (1 - 1/a(n)) = A132026. (End)

A309908 a(n) is 2^n represented in bijective base-9 numeration.

Original entry on oeis.org

1, 2, 4, 8, 17, 35, 71, 152, 314, 628, 1357, 2725, 5551, 12212, 24424, 48848, 98797, 218715, 438531, 878162, 1867334, 3845668, 7792447, 16694895, 34499911, 69121922, 149243944, 299487988, 619987187, 1342185385, 2684381781, 5478773672, 11968657454, 24148425918
Offset: 0

Views

Author

Alois P. Heinz, Aug 21 2019

Keywords

Comments

Differs from A001357 first at n = 16: a(16) = 98797 < 108807 = A001357(16).

Examples

			a(10) =  1357_bij9 =       9*(9*(9*1+3)+5)+7 =  1024 = 2^10.
a(16) = 98797_bij9 = 9*(9*(9*(9*9+8)+7)+9)+7 = 65536 = 2^16.

Links

Crossrefs

Cf. A000079, A001357, A047855, A052382, A093135, A214676, A325203.

Programs

  • Maple
    b:= proc(n) local d, l, m; m:= n; l:= "";
      while m>0 do d:= irem(m, 9, 'm');
        if d=0 then d:=9; m:= m-1 fi; l:= d, l
      od; parse(cat(l))
    end:
a:= n-> b(2^n):
seq(a(n), n=0..33);

Formula

a(n) = A052382(2^n) = A052382(A000079(n)).

A173736 a(n) = (10^n + 35)/9 for n>0.

Original entry on oeis.org

5, 15, 115, 1115, 11115, 111115, 1111115, 11111115, 111111115, 1111111115, 11111111115, 111111111115, 1111111111115, 11111111111115, 111111111111115, 1111111111111115, 11111111111111115, 111111111111111115, 1111111111111111115, 11111111111111111115, 111111111111111111115
Offset: 1

Views

Author

Vincenzo Librandi, Feb 23 2010

Keywords

Links

Crossrefs

Cf. A093135.

Programs

  • Magma
    [(10^n+35)/9: n in [1..20]]; // Vincenzo Librandi, Jul 05 2012
  • Mathematica
    CoefficientList[Series[(5-40*x)/((1-x)*(1-10*x)),{x,0,30}],x] (* Vincenzo Librandi, Jul 05 2012 *)

Formula

a(n) = a(n-1) + 10^(n-1) = 10*a(n-1) - 35 with n>0, a(0)=4.
From Vincenzo Librandi, Jul 05 2012: (Start)
G.f.: 5*x*(1-8*x)/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2). (End)
From Elmo R. Oliveira, Jun 18 2025: (Start)
E.g.f.: -4 + exp(x)*(35 + exp(9*x))/9.
a(n) = 5*A093135(n-1). (End)
Showing 1-3 of 3 results.

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