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.

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A144788 a(n+1) = a(n)^3 - a(n) + 1 with a(1) = 3.

Original entry on oeis.org

3, 25, 15601, 3797146111201, 54748462363349863789331489199516542401, 164102719777453765093060237305105241279744702931788833463132240094953276902575203916973404664183940277497050364801
Offset: 1

Views

Author

Artur Jasinski, Sep 22 2008

Keywords

Crossrefs

Cf. A000058, A082732, A144779-A144788.

Programs

  • Mathematica
    a = {}; k = 3; Do[AppendTo[a, k]; k = k^3 - k + 1, {n, 1, 8}]; a (* Artur Jasinski *)
NestList[#^3-#+1&,3,5] (* Harvey P. Dale, Jan 09 2014 *)

Formula

a(n) = round(c^(3^n)), where c = 1.4298877386... is given in A144811.

A144787 Recurrence sequence a(n+1)=a(n)^3-a(n)+1 and a(1)=2.

Original entry on oeis.org

2, 7, 337, 38272417, 56060590716839257663297, 176186654453940966415101758343368831005891099500239113100063334235777
Offset: 1

Views

Author

Artur Jasinski, Sep 22 2008

Keywords

Comments

For constant c=1.240554576397679299452... see A144810.

Crossrefs

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

Programs

  • Mathematica
    a = {}; k = 2; Do[AppendTo[a, k]; k = k^3 - k + 1, {n, 1, 8}]; a (*Artur Jasinski*)
NestList[#^3-#+1&,2,5] (* Harvey P. Dale, Jun 24 2013 *)

A004168 a(n+1) = a(n)*(a(n)+1).

Original entry on oeis.org

3, 12, 156, 24492, 599882556, 359859081592975692, 129498558604939936868397356895854556, 16769876680757063368089314196389622249367851612542961252860614401811692
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The next term (a(8)) has 141 digits. - Harvey P. Dale, Jul 02 2021

Crossrefs

Cf. A000058, A007018.

Programs

  • Magma
    [n eq 1 select 3 else  Self(n-1)*(Self(n-1)+1): n in [1..10]]; // Vincenzo Librandi, Feb 23 2016
  • Maple
    A004168 := proc(n) option remember; if n=0 then 3 else A004168(n-1)*(A004168(n-1)+1); fi; end;
  • Mathematica
    a = {3}; Do[AppendTo[a, a[[n - 1]] (a[[n - 1]] + 1)], {n, 2, 8}]; a (* Michael De Vlieger, Feb 23 2016 *)
NestList[#(#+1)&,3,7] (* Harvey P. Dale, Jul 02 2021 *)

Formula

a(n) = A082732(n+3) - 1. - Max Alekseyev, Aug 09 2019

Extensions

a(7) from Vincenzo Librandi, Feb 23 2016

A144743 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=3.

Original entry on oeis.org

3, 5, 19, 341, 115939, 13441735781, 180680260792773944179, 32645356640144805339284259388335434039861, 1065719310162246533488642668727242229836148490441005113524301742665845135502859459
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.
Conjecture: A130282 and this sequence are disjoint. If this is true, for n >= 1, a(n+1) is the smallest m such that (m^2-1) / (a(n)^2-1) + 1 is a square. - Jianing Song, Mar 19 2022

Crossrefs

Cf. A000058, A082732, A144744, A144745, A144746, A144747, A144748.

Programs

  • Mathematica
    a = {3}; k = 3; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a
  • PARI
    a(n,s=3)=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, a(0)=3.
a(n) ~ c^(2^n), where c = 2.07259396780115004655284076205241023281287049774423620992171834046728756... . - Vaclav Kotesovec, May 06 2015

Extensions

Edited by M. F. Hasler, Oct 06 2014

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

A376062 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence {7/6, 5/4, 5/4, 5/4, ...}.

Original entry on oeis.org

2, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693
Offset: 1

Views

Author

N. J. A. Sloane, Sep 14 2024

Keywords

Comments

This sequence and A376186 were discovered by Rémy Sigrist on Sep 09 2024. The two sequences {b(1)=7/6, b(k)=5/4 for k>1} and {b(1)=5/4, b(2*k)=3/2, b(2*k+1)=6/5 for k>0} are the first sequences {b(i)} discovered with the property that the sums S(n) do not converge to numbers of the form (e_n - 1)/e_n as n-> oo.
This is essentially the same sequence as A004168 and A082732.

Links

Crossrefs

Cf. A004168, A082732, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376061, A376185.

Programs

  • Mathematica
    Join[{2}, RecurrenceTable[{a[n+1] == a[n]^2 - a[n] + 1, a[2] == 4}, a, {n, 2, 9}]] (* Amiram Eldar, Sep 15 2024 *)

Formula

a(n+1) = a(n)^2 - a(n) + 1 for n >= 2.
Previous Showing 11-20 of 24 results. Next

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

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