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 1-4 of 4 results.

A172162 a(n) = ( A165154(n) + A165155(n) )/2.

Original entry on oeis.org

0, 1, 101, 10201, 1020401, 102050701, 10205121101, 1020513261601, 102051333512201, 10205133479922901, 1020513348977553701, 102051334912467474601, 10205133491373712765601, 1020513349139081705516701, 102051334913924160974827901, 10205133491392617410795809201
Offset: 0

Views

Author

Mark Dols, Jan 27 2010

Keywords

Crossrefs

Programs

  • Mathematica
    Table[99 100^n/9701 - 11^n/178 - (-9)^n/218, {n, 0, 20}] (* Bruno Berselli, Oct 02 2015 *)
  • PARI
    concat(0, Vec(-x*(x-1)/((9*x+1)*(11*x-1)*(100*x-1)) + O(x^30))) \\ Colin Barker, Oct 02 2015
    
  • SageMath
    [(-89*(-9)^n - 109*11^n + 198*10^(2*n))/19402 for n in (0..50)] # G. C. Greubel, Apr 24 2022

Formula

a(n) = 99*100^n/9701 - 11^n/178 - (-9)^n/218. [Bruno Berselli, Oct 02 2015]
From Colin Barker, Oct 02 2015: (Start)
a(n) = 102*a(n-1) - 101*a(n-2) - 9900*a(n-3) for n>3.
G.f.: x*(1-x) / ((1+9*x)*(1-11*x)*(1-100*x)).
(End)

Extensions

a(0) and more terms added by Bruno Berselli, Oct 02 2015

A172163 a(n) = ( A165155(n) - A165154(n) )/2.

Original entry on oeis.org

0, 0, 10, 1020, 103030, 10307040, 1030814050, 103082025060, 10308214641070, 1030821549763080, 103082156348992090, 10308215646124529100, 1030821564770799275110, 103082156478507926931120, 10308215647869324982098130, 1030821564787110934730377140
Offset: 0

Views

Author

Mark Dols, Jan 27 2010

Keywords

Crossrefs

Programs

  • Mathematica
    Table[10^(2 n + 1)/9701 - 11^n/178 + (-9)^n/218, {n, 0, 20}] (* Bruno Berselli, Oct 02 2015 *)
    LinearRecurrence[{102,-101,-9900},{0,0,10},20] (* Harvey P. Dale, Aug 17 2021 *)
  • PARI
    concat([0,0], Vec(10*x^2/((9*x+1)*(11*x-1)*(100*x-1)) + O(x^30))) \\ Colin Barker, Oct 02 2015
    
  • SageMath
    [(89*(-9)^n + 2*10^(2*n+1) - 109*11^n)/19402 for n in (0..50)] # G. C. Greubel, Apr 24 2022

Formula

a(n) = 10^(2*n+1)/9701 - 11^n/178 + (-9)^n/218. [Bruno Berselli, Oct 02 2015]
From Colin Barker, Oct 02 2015: (Start)
a(n) = 102*a(n-1) - 101*a(n-2) - 9900*a(n-3) for n>2.
G.f.: 10*x^2 / ((1+9*x)*(1-11*x)*(1-100*x)).
(End)

Extensions

a(0)=0 and more terms added by Bruno Berselli, Oct 02 2015

A165154 a(n) = 100*a(n-1) + (-9)^(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 1, 91, 9181, 917371, 91743661, 9174307051, 917431236541, 91743118871131, 9174311930159821, 917431192628561611, 91743119266342945501, 9174311926602913490491, 917431192660573778585581, 91743119266054835992729771, 9174311926605506476065432061
Offset: 0

Views

Author

Mark Dols, Sep 05 2009

Keywords

Crossrefs

Programs

  • Magma
    [(1/109)*(100^n-(-9)^n): n in [0..20]]; // Vincenzo Librandi, Jun 10 2011
    
  • Mathematica
    LinearRecurrence[{91,900}, {0,1}, 40] (* G. C. Greubel, Feb 09 2023 *)
  • PARI
    Vec(x/((1+9*x)*(1-100*x)) + O(x^20)) \\ Colin Barker, Oct 02 2015
    
  • SageMath
    [(100^n-(-9)^n)/109 for n in range(41)] # G. C. Greubel, Feb 09 2023

Formula

From Colin Barker, Oct 02 2015: (Start)
a(n) = 91*a(n-1) + 900*a(n-2) for n>1, a(0)=0.
G.f.: x/((1+9*x)*(1-100*x)). (End)
E.g.f.: (1/109)*(exp(100*x) - exp(-9*x)). - G. C. Greubel, Feb 09 2023

Extensions

a(0) prepended by Joerg Arndt, Oct 02 2015

A172175 a(n) = 110*a(n-1) + 1.

Original entry on oeis.org

1, 111, 12211, 1343211, 147753211, 16252853211, 1787813853211, 196659523853211, 21632547623853211, 2379580238623853211, 261753826248623853211, 28792920887348623853211, 3167221297608348623853211, 348394342736918348623853211, 38323377701061018348623853211
Offset: 1

Views

Author

Mark Dols, Jan 28 2010

Keywords

Comments

Sum of pairs of integers given in A162849. Sum of digits give A000225.

Crossrefs

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, 110 a[[n-1]] + 1], {n, 2, 15}]; a (* Michael De Vlieger, Oct 02 2015 *)
  • PARI
    Vec(1/((x-1)*(110*x-1)) + O(x^30)) \\ Colin Barker, Oct 02 2015
    
  • SageMath
    [((110)^n -1)/109 for n in (1..50)] # G. C. Greubel, Apr 26 2022

Formula

From Colin Barker, Oct 02 2015: (Start)
a(n) = 111*a(n-1) - 110*a(n-2) for n>2.
G.f.: 1 / ((1-x)*(1-110*x)). (End)
Showing 1-4 of 4 results.