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 1-6 of 6 results.

A082732 a(1) = 1, a(2) = 3, a(n) = LCM of all the previous terms + 1.

Original entry on oeis.org

1, 3, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693
Offset: 1

Views

Author

Amarnath Murthy, Apr 14 2003

Keywords

Comments

The LCM is in fact the product of all previous terms. From a(5) onwards the terms alternately end in 57 and 93.

Crossrefs

Cf. A000058, A004168, A144743, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

Programs

  • Mathematica
    a[1] = 1; a[2] = 3; a[n_] := Apply[LCM, Table[a[i], {i, 1, n - 1}]] + 1; Table[ a[n], {n, 1, 10}]
c=1.8806785436830780944921917650127503562630617563236301969047995953391479871\
7695395204087358090874194124503892563356447954254847544689332763; Table[c^(2^n),{n,1,6}] or a = {}; k = 4; Do[AppendTo[a, k]; k = k^2 - k + 1, {n, 1, 10}]; a (* Artur Jasinski, Sep 22 2008 *)

Formula

For n>=3, a(n+1) = a(n)^2 - a(n) + 1.
For n>=3, a(n) = A004168(n-3) + 1. - Max Alekseyev, Aug 09 2019
1/3 = Sum_{n=3..oo} 1/a(n) = 1/4 + 1/13 + 1/157 + 1/24493 + ... or 1 = Sum_{n=3..oo} 3/a(n) = 3/4 + 3/13 + 3/157 + 3/24493 + .... If we take segment of length 1 and cut off in each step fragment of maximal length such that numerator of fraction is 3, denominators of such fractions will be successive numbers of this sequence. - Artur Jasinski, Sep 22 2008
a(n+2)=1.8806785436830780944921917650127503562630617563236301969047995953391\
4798717695395204087358090874194124503892563356447954254847544689332763...^(2^n). - Artur Jasinski, Sep 22 2008

Extensions

More terms from Robert G. Wilson v, Apr 15 2003

A144744 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=4.

Original entry on oeis.org

4, 11, 109, 11771, 138544669, 19194625169774891, 368433635408155743950638444286989, 135743343700069833946317076518699443524748244656296738254150399131
Offset: 0

Views

Author

Artur Jasinski, Sep 20 2008

Keywords

Comments

a(0)=3 is the smallest integer generating an increasing sequence of the form a(n)=a(n-1)^2-a(n-1)-1.

Crossrefs

Cf. A000058, A082732, A144743, A144745, A144746, A144747, A144748.

Programs

  • Mathematica
    a = {}; k = 4; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a
  • PARI
    a(n, s=4)={for(i=1, n, s=s^2-s-1); s} \\ M. F. Hasler, Oct 06 2014

Formula

a(n)=a(n-1)^2-a(n-1)-1 and a(0)=4.
a(n) ~ c^(2^n), where c = 3.22737450272053234771396610986262048906046050824600724014923334412606964... . - Vaclav Kotesovec, May 06 2015

Extensions

Edited by M. F. Hasler, Oct 06 2014

A144745 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=9.

Original entry on oeis.org

9, 71, 4969, 24685991, 609398126966089, 371366077149776919833628989831, 137912763257614063309949706968500684963726537144819872418729
Offset: 0

Views

Author

Artur Jasinski, Sep 20 2008

Keywords

Comments

The original version of this sequence had a(0)=5=A144743(1) and therefore was essentially the same as that sequence A144743.
The next term a(8) has 119 digits.

Crossrefs

Cf. A000058, A082732, A144743, A144744, A144746, A144747, A144748.

Programs

  • Mathematica
    k = 9; a = {k}; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a
NestList[#^2 - # - 1 &, 9, 7]  (* Harvey P. Dale, Feb 04 2011 *)
  • PARI
    a(n,s=9)=for(i=1,n,s=s^2-s-1);s \\ M. F. Hasler, Oct 06 2014

Formula

a(n) = a(n-1)^2-a(n-1)-1 and a(0)=9.
a(n) ~ c^(2^n), where c = 8.395688554881795978328174160925857176207363473280394010762212170489... . - Vaclav Kotesovec, May 06 2015

Extensions

New initial value a(0)=9 from M. F. Hasler, Oct 20 2014

A144746 a(n) = a(n-1)^2 - a(n-1) - 1, a(0)=6.

Original entry on oeis.org

6, 29, 811, 656909, 431528777371, 186217085698878552894269, 34676803006183479266409218250231853558140150091, 1202480666729655584789949373132702064208272454072740050128160074167965751208292536045867158189
Offset: 0

Views

Author

Artur Jasinski, Sep 20 2008

Keywords

Comments

a(0)=3 is the smallest integer generating an increasing sequence of the form a(n) = a(n-1)^2 - a(n-1) - 1, cf. A144743.

Crossrefs

Cf. A000058, A082732, A144743, A144744, A144745, A144747, A144748.

Programs

  • Mathematica
    NestList[#^2-#-1&,6,8]  (* Harvey P. Dale, Jan 22 2011 *)
  • PARI
    a(n, s=6)={for(i=1, n, s=s^2-s-1);s} \\ M. F. Hasler, Oct 06 2014

Formula

a(n) = a(n-1)^2 - a(n-1) - 1 and a(0)=6.
a(n) ~ c^(2^n), where c = 5.33565954034691307256446890777476398311129407641143635105306409567572... . - Vaclav Kotesovec, May 06 2015

Extensions

Corrected and edited by M. F. Hasler, Oct 06 2014

A144747 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=7.

Original entry on oeis.org

7, 41, 1639, 2684681, 7207509387079, 51948191564824694742765161, 2698614606855723567054656642857156538246857652590759, 7282520796335071470236496456671241855257664867148949932302276253455702665493855273950765616767079605321
Offset: 0

Views

Author

Artur Jasinski, Sep 20 2008

Keywords

Comments

a(0)=3 is the smallest integer generating an increasing sequence of the form a(n)=a(n-1)^2-a(n-1)-1, cf. A144743.

Crossrefs

Cf. A000058, A082732, A144743, A144744, A144745, A144746, A144748.

Programs

  • Mathematica
    a = {}; k = 7; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a
  • PARI
    a(n, s=7)={for(i=1, n, s=s^2-s-1);s} \\ M. F. Hasler, Oct 06 2014

Formula

a(n)=a(n-1)^2-a(n-1)-1 and a(0)=7.
a(n) ~ c^(2^n), where c = 6.3622623884585267364822329679498420997632627444610172910703030892754... . - Vaclav Kotesovec, May 06 2015

Extensions

Edited by M. F. Hasler, Oct 06 2014

A144748 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=8.

Original entry on oeis.org

8, 55, 2969, 8811991, 77651176572089, 6029705223029665929437251831, 36357345076631233348346773693633697407708655232275600729, 1321856541021241383115043586121503961331042183698683965174269952435581223368633124721267107619465028785549730711
Offset: 0

Views

Author

Artur Jasinski, Sep 20 2008

Keywords

Comments

a(0)=3 is the smallest integer generating an increasing sequence of the form a(n)=a(n-1)^2-a(n-1)-1, cf. A144743.

Crossrefs

Cf. A000058, A082732, A144743, A144744, A144745, A144746, A144747.

Programs

  • Mathematica
    a = {}; k = 8; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a
NestList[#^2-#-1&,8,10] (* Harvey P. Dale, Mar 14 2016 *)
  • PARI
    a(n, s=8)={for(i=1, n, s=s^2-s-1); s} \\ M. F. Hasler, Oct 06 2014

Formula

a(n)=a(n-1)^2-a(n-1)-1 and a(0)=8.
a(n) ~ c^(2^n), where c = 7.3813237216360344087566795911708086794628396333350474334044779783264... . - Vaclav Kotesovec, May 06 2015

Extensions

Edited by M. F. Hasler, Oct 06 2014
Showing 1-6 of 6 results.

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