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 1-7 of 7 results.

A198693 a(n) = 3*4^n-1.

Original entry on oeis.org

2, 11, 47, 191, 767, 3071, 12287, 49151, 196607, 786431, 3145727, 12582911, 50331647, 201326591, 805306367, 3221225471, 12884901887, 51539607551, 206158430207, 824633720831, 3298534883327, 13194139533311, 52776558133247
Offset: 0

Views

Author

Vincenzo Librandi, Oct 29 2011

Keywords

Links

Crossrefs

Cf. A024036, A097743, A114569, A156760.

Programs

  • Magma
    [3*4^n-1: n in [0..30]]
  • Mathematica
    3*4^Range[0,30]-1 (* or *) LinearRecurrence[{5,-4},{2,11},30] (* Harvey P. Dale, Jul 04 2017 *)

Formula

a(n) = 4*a(n-1)+3.
a(n) = 5*a(n-1) - 4*a(n-2), for n > 1.
G.f.: (2+x)/((4*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011

A098102 a(n) = 2^(prime(n) - 1) - 1 where prime(n) is the n-th prime.

Original entry on oeis.org

1, 3, 15, 63, 1023, 4095, 65535, 262143, 4194303, 268435455, 1073741823, 68719476735, 1099511627775, 4398046511103, 70368744177663, 4503599627370495, 288230376151711743, 1152921504606846975
Offset: 1

Views

Author

Parthasarathy Nambi, Sep 22 2004

Keywords

Examples

			For n = 3; prime(3) = 5, a(3) = 2^(5-1) - 1 = 15.

Crossrefs

Cf. A097743, A000225.

Programs

  • Magma
    [2^(NthPrime(n)-1) - 1: n in [1..18]]; // Jaroslav Krizek, Jan 02 2015
  • Mathematica
    Table[2^(Prime[n] - 1) - 1, {n, 20}]
  • PARI
    a(n) = 2^(prime(n)-1) - 1; \\ Michel Marcus, Jan 13 2023

Extensions

Edited and extended by Robert G. Wilson v, Sep 23 2004
Definition and example corrected by Jaroslav Krizek, Jan 02 2015
Name edited by Michel Marcus, Jan 13 2023

A114569 a(n) = 9*4^n - 1.

Original entry on oeis.org

8, 35, 143, 575, 2303, 9215, 36863, 147455, 589823, 2359295, 9437183, 37748735, 150994943, 603979775, 2415919103, 9663676415, 38654705663, 154618822655, 618475290623, 2473901162495, 9895604649983, 39582418599935, 158329674399743, 633318697598975, 2533274790395903
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)excite.com), Feb 16 2006

Keywords

Comments

Squares of the cotangents of the arcsins of 1/(3*2^n).

Examples

			a(2) = 143.

Links

Crossrefs

Cf. A024036, A097743, A156760, A199208.

Programs

Formula

From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 4*a(n-1) + 3, n>0; a(0)=8.
a(n) = 5*a(n-1) - 4*a(n-2), n>1; a(0)=8, a(1)=35.
G.f.: (8-5*x)/(1-5*x+4*x^2). (End)
From Elmo R. Oliveira, May 08 2025: (Start)
E.g.f.: exp(x)*(9*exp(3*x) - 1).
a(n) = A199208(n) - 2. (End)

Extensions

More terms from Stefan Steinerberger, Feb 16 2006

A198694 a(n) = 7*4^n-1.

Original entry on oeis.org

6, 27, 111, 447, 1791, 7167, 28671, 114687, 458751, 1835007, 7340031, 29360127, 117440511, 469762047, 1879048191, 7516192767, 30064771071, 120259084287, 481036337151, 1924145348607, 7696581394431, 30786325577727, 123145302310911, 492581209243647, 1970324836974591
Offset: 0

Views

Author

Vincenzo Librandi, Oct 29 2011

Keywords

Links

Crossrefs

Cf. A024036, A097743, A114569, A156760.

Programs

  • Magma
    [7*4^n-1: n in [0..30]]
  • Mathematica
    7*4^Range[0,30]-1 (* or *) LinearRecurrence[{5,-4},{6,27},30] (* Harvey P. Dale, Nov 14 2018 *)

Formula

a(n) = 4*a(n-1)+3.
a(n) = 5*a(n-1)-4*a(n-2), n>1.
G.f.: ( 6-3*x ) / ( (4*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011
E.g.f.: exp(x)*(7*exp(3*x) - 1). - Stefano Spezia, Apr 17 2024

A198695 a(n) = 11*4^n - 1.

Original entry on oeis.org

10, 43, 175, 703, 2815, 11263, 45055, 180223, 720895, 2883583, 11534335, 46137343, 184549375, 738197503, 2952790015, 11811160063, 47244640255, 188978561023, 755914244095, 3023656976383, 12094627905535, 48378511622143, 193514046488575, 774056185954303, 3096224743817215
Offset: 0

Views

Author

Vincenzo Librandi, Oct 29 2011

Keywords

Links

Crossrefs

Cf. A024056, A097743, A114569, A156760, A199211.

Programs

  • Magma
    [11*4^n-1: n in [0..30]];
  • Mathematica
    11*4^Range[0,30]-1 (* or *) NestList[4#+3&,10,30] (* or *) LinearRecurrence[ {5,-4},{10,43},30] (* Harvey P. Dale, Aug 07 2021 *)

Formula

a(n) = 4*a(n-1) + 3.
a(n) = 5*a(n-1) - 4*a(n-2), n > 1.
G.f.: (10-7*x)/((4*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011
From Elmo R. Oliveira, May 07 2025: (Start)
E.g.f.: exp(x)*(11*exp(3*x) - 1).
a(n) = A199211(n) - 2. (End)

A098116 a(n) = 3^(p-1) + (3^p - 1) where p is the n-th prime.

Original entry on oeis.org

11, 35, 323, 2915, 236195, 2125763, 172186883, 1549681955, 125524238435, 91507169819843, 823564528378595, 600378541187996483, 48630661836227715203, 437675956526049436835, 35451752478610004383715
Offset: 1

Views

Author

Parthasarathy Nambi, Sep 23 2004

Keywords

Examples

			a(1) = 3^(2-1) + 3^2 - 1 = 11.

Crossrefs

Cf. A097743.

Programs

  • Mathematica
    Table[3^(Prime[n] - 1) + (3^Prime[n] - 1), {n, 1, 20}] (* Stefan Steinerberger, Feb 28 2006 *)

Extensions

More terms from Stefan Steinerberger, Feb 28 2006

A098117 a(n) = 5^(prime(n) - 1) + 5^prime(n) - 1.

Original entry on oeis.org

29, 149, 3749, 93749, 58593749, 1464843749, 915527343749, 22888183593749, 14305114746093749, 223517417907714843749, 5587935447692871093749, 87311491370201110839843749, 54569682106375694274902343749, 1364242052659392356872558593749
Offset: 1

Views

Author

Parthasarathy Nambi, Sep 23 2004

Keywords

Examples

			a(1) = 5^(2 - 1) + (5^2 - 1) = 29.

Crossrefs

Cf. A097743.

Programs

  • Magma
    [5^(NthPrime(n)-1) + 5^NthPrime(n) - 1: n in [1..20]]; // Vincenzo Librandi, Aug 27 2015
  • Mathematica
    Table[5^(Prime[n] - 1) + (5^Prime[n] - 1), {n, 1, 15}] (* Stefan Steinerberger, Feb 28 2006 *)
  • PARI
    main(m)=forprime(p=2,100, print1(", ",5^(p-1) + (5^p - 1))) \\ Anders Hellström, Aug 26 2015

Formula

a(n) = A198764(prime(n)-1) = A198764(A000040(n)-1). - Michel Marcus, Aug 27 2015

Extensions

More terms from Stefan Steinerberger, Feb 28 2006
a(14) from Vincenzo Librandi, Aug 27 2015
Showing 1-7 of 7 results.

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