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 1-5 of 5 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

Links

Crossrefs

Cf. A162741, A162849, A165154, A165155, A172163.

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

Links

Crossrefs

Cf. A162741, A162849, A165154, A165155, A172162.

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

A161999 For n even a(n) = a(n-1) + 10*a(n-2), for n odd a(n) = a(n-3) + 10 a(n-2); with a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 1, 10, 20, 101, 301, 1030, 4040, 10601, 51001, 110050, 620060, 1151501, 7352101, 12135070, 85656080, 128702801, 985263601, 1372684090, 11225320100, 14712104501, 126965305501, 158346365110, 1427999420120
Offset: 1

Views

Author

Mark Dols, Jun 24 2009, Jun 28 2009, Jul 13 2009

Keywords

Examples

			As pairs:
0, 1
1, 10
20, 101
301, 1030
4040, 10601
51001, 110050
620060, 1151501
7352101, 12135070
85656080, 128702801

Links

Crossrefs

Combination of A081192 and A016190. Triangle A007318 (even /uneven rows). Partly same function as A015446. A001020 (as sum of pairs of 2n).A001019 (as difference of pairs of 2n)
Cf. A162849.

Programs

  • Mathematica
    nxt[{n_,a_,b_,c_}]:={n+1,b,c,If[OddQ[n],c+10b,a+10b]}; NestList[nxt,{2,0,1,1},30][[All,2]] (* or *) LinearRecurrence[{0,20,0,-99},{0,1,1,10},30] (* Harvey P. Dale, May 03 2018 *)

Formula

a(n)=20*a(n-2)-99*a(n-4). G.f.: -x^2*(-1-x+10*x^2)/((3*x-1)*(3*x+1)*(11*x^2-1)). [From R. J. Mathar, Jul 13 2009]

Extensions

Edited by N. J. A. Sloane, Jun 30 2009
NAME adapted to offset. - R. J. Mathar, Jun 19 2021

A172174 a(n) = 90*a(n-1) + 1.

Original entry on oeis.org

1, 91, 8191, 737191, 66347191, 5971247191, 537412247191, 48367102247191, 4353039202247191, 391773528202247191, 35259617538202247191, 3173365578438202247191, 285602902059438202247191, 25704261185349438202247191, 2313383506681449438202247191
Offset: 1

Views

Author

Mark Dols, Jan 28 2010

Keywords

Comments

Difference of pairs of integers given in A162849.
Sum of digits give A017173.

Links

Crossrefs

Cf. A017173, A162849, A165154.

Programs

  • Mathematica
    NestList[90#+1&,1,20] (* Harvey P. Dale, Aug 29 2014 *)
  • PARI
    Vec(1/((x-1)*(90*x-1)) + O(x^30)) \\ Colin Barker, Oct 02 2015
  • SageMath
    [(90^n -1)/89 for n in (1..50)] # G. C. Greubel, Apr 26 2022

Formula

From Colin Barker, Oct 02 2015: (Start)
a(n) = 91*a(n-1) - 90*a(n-2) for n>2.
G.f.: 1 / ((1-x)*(1-90*x)). (End)

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.

Links

Crossrefs

Cf. A000225, A162849, A165155.

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 1-5 of 5 results.

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

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