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.

A064526 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n).

Original entry on oeis.org

0, 1, 2, 3, 5, 13, 49, 529, 21121, 10369921, 213952189441, 2214253468601687041, 473721461635593679669210030081, 1048939288228833101089604217183056027094304481281
Offset: 0

Views

Author

Michael Somos, Oct 07 2001

Keywords

Comments

Every nonzero term is relatively prime to all others (which proves that there are infinitely many primes). See A236394 for the primes that appear.

Links

Crossrefs

Cf. A001685, A003686, A064183, A064847, A070231, A070233, A070234, A094303.
See A236394 for the primes that are produced.

Programs

  • Mathematica
    Flatten[{0,1, RecurrenceTable[{a[n]==(a[n-1]^2 + a[n-2]^2 - a[n-1]*a[n-2] * (1+a[n-2]))/(1-a[n-2]), a[2]==2, a[3]==3},a,{n,2,15}]}] (* Vaclav Kotesovec, May 21 2015 *)
  • PARI
    {a(n) = local(v); if( n<3, max(0, n), v = [1,1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[1] + v[2])}
  • PARI
    {a(n) = if( n<4, max(0, n), (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)))}

Formula

a(n) = (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)) for n >= 2.
a(n) ~ c^(phi^n), where c = 1.2364241784241086061606568429916822975882631646194967549068405592472125928485... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

A070231 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k) for k >= 1; then a(n) = u(n).

Original entry on oeis.org

1, 3, 7, 31, 1279, 9202687, 3692849258577919, 98367959484921734629696721986125823, 3882894052327310957045599009145809243674851356642054390303168725061781159935999
Offset: 1

Views

Author

Benoit Cloitre, May 08 2002

Keywords

Links

Crossrefs

Cf. A003686, A064183, A064526, A064847, A070233 (= w), A070234 (= v), A094303, A160389, A255249.

Programs

  • Mathematica
    a[1] = 1; v[1] = 1; w[1] = 1; a[k_] := a[k] = a[k - 1] + v[k - 1] + w[k - 1]; v[k_] := v[k] = a[k - 1]*v[k - 1] + v[k - 1]*w[k - 1] + w[k - 1]*a[k - 1]; w[k_] := w[k] = a[k - 1]*v[k - 1]*w[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)
  • PARI
    lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1];); u; } \\ Petros Hadjicostas, May 11 2020

Formula

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0; that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n)= Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]
a(n) ~ gu^((1 + C)^n), where C is defined above and gu = 1.131945853718244297... The relation between constants gu, gv (see A070234) and gw (see A070233) is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A070233 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k) for k >= 1; then a(n) = w(n).

Original entry on oeis.org

1, 1, 9, 945, 8876385, 3689952451492545, 98367948795841301790914258556831105, 3882894052327309905582682317031276840071039865528864289025562807872336355445505
Offset: 1

Views

Author

Benoit Cloitre, May 08 2002

Keywords

Comments

Next term is too large to include.

Links

Crossrefs

Cf. A003686, A064183, A064526, A064847, A070231 (= u), A070234 (= v), A094303, A160389, A255249.

Programs

  • Mathematica
    u[1] = 1; v[1] = 1; a[1] = 1; u[k_] := u[k] = u[k - 1] + v[k - 1] + a[k - 1]; v[k_] := v[k] = u[k - 1]*v[k - 1] + v[k - 1]*a[k - 1] + a[k - 1]*u[k - 1]; a[k_] := a[k] = u[k - 1]*v[k - 1]*a[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)
  • PARI
    lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1]; ); w; } \\ Petros Hadjicostas, May 11 2020

Formula

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0: that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n)= Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]
a(n) ~ gw^((C + 1)^n), where C is defined above and gw = 1.321128752475732548... The relation between constants gu (see A070231), gv (see A070234) and gw is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A070234 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k); then a(n) = v(n).

Original entry on oeis.org

1, 3, 15, 303, 325023, 2896797882687, 10689080432835089614170716799, 1051462916692114532403603811392745230616355871287492722818364671
Offset: 1

Views

Author

Benoit Cloitre, May 08 2002

Keywords

Links

Crossrefs

Cf. A003686, A064183, A064526, A064847, A070231 (= u), A070233 (= w), A094303, A160389, A255249.

Programs

Formula

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0; that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n) = Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]
a(n) ~ gv^((C + 1)^n), where C is defined above and gv = 1.250231610564761084... The relation between constants gu (see A070231), gv and gw (see A070233) is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A094303 a(1) = 1, a(2) = 2, and a(n+1) = a(n) * sum of all previous terms up to a(n-1) for n >= 2.

Original entry on oeis.org

1, 2, 2, 6, 30, 330, 13530, 5019630, 69777876630, 351229105131280530, 24509789089304573335878465330, 8608552999157278550998626549630446732052243030
Offset: 1

Views

Author

Amarnath Murthy, Apr 29 2004

Keywords

Comments

From Petros Hadjicostas, May 11 2020: (Start)
R. J. Mathar's conjecture is correct and this is identical to A064847 starting at n = 3. To see why this is the case, consider the sequences u(n) and v(n) defined by u(1) = v(1) = 1, and u(k+1) = u(k) + v(k), v(k+1) = u(k)*v(k) for k >= 1. Then u(n) = A003686(n) and v(n) = A064847(n) for n >= 1.
Then v(n) = u(n+1) - u(n), and thus Sum_{k=1..n-1} v(k) = u(n) - u(1) = u(n) - 1 for n >= 2. Then v(n-1) + ... + v(3) + (v(2) + 1) + v(1) = u(n) for n >= 3, and hence v(n)*(v(n-1) + ... + v(3) + (v(2) + 1) + v(1)) = u(n)*v(n) = v(n+1).
Since v(1) = 1 = a(1) and v(2) + 1 = 2 = a(2), the sequence (v(1), v(2) + 1, v(3), ..., v(n), ...) is identical to the current sequence. Hence, a(n) = v(n) = u(n+1) - u(n) = A003686(n+1) - A003686(n) for n >= 3. (End)

Links

Crossrefs

See A064847 for another version.
Cf. A003686, A064183, A064526, A070231, A070233, A070234.

Programs

  • Mathematica
    nxt[{t1_,t2_,a_}]:=Module[{c=t1*a},{t1+t2,c,c}]; Join[{1},NestList[nxt,{1,2,2},10][[All,2]]] (* Harvey P. Dale, Aug 30 2020 *)
  • PARI
    lista(nn) = { my(va = vector(nn)); va[1] = 1; va[2] = 2; for(n=3, nn, va[n] = va[n-1]*sum(k=1, n-2, va[k]);); va; } \\ Petros Hadjicostas, May 11 2020

Formula

Conjecture: a(n) = A003686(n+1) - A003686(n) for n >= 3. - R. J. Mathar, Apr 24 2007

Extensions

More terms from Gareth McCaughan, Jun 10 2004

A236394 Primes produced by A064526, in order of appearance.

Original entry on oeis.org

2, 3, 5, 13, 7, 23, 21121, 853, 12157, 213952189441, 31, 71427531245215711, 17, 163, 68743, 28031803, 88717035481559, 4549, 38197, 3835420378661, 1573954557128833179852821957
Offset: 1

Views

Author

N. J. A. Sloane, Jan 28 2014

Keywords

Crossrefs

Cf. A064526, A064183.

A126023 a(0)=0, a(1)=1; for n>1, a(n) = a(n-1)*(a(n-1)+a(n-2)).

Original entry on oeis.org

0, 1, 1, 2, 6, 48, 2592, 6842880, 46842743439360, 2194242933464976548324966400, 4814702051061088283920560140388303599459408453566464000, 23181355840491850372772514246989811472332466216882815765831029699284672633019505150499832539732598430105600000
Offset: 0

Views

Author

Franklin T. Adams-Watters, Feb 27 2007

Keywords

Links

Crossrefs

Cf. A064183.

Programs

  • Mathematica
    a=0;b=1;lst={a,b};Do[c=(a+b)*b;AppendTo[lst,c];a=b;b=c,{n,2*3!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n] == a[n-1]*(a[n-1]+a[n-2])}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
  • PARI
    a=0;b=1;vector(11,i,c=b*(b+a);a=b;b=c;a)

Formula

a(n) ~ c^(2^n), where c = 1.130839439573188327984771046199629891088102083459871373699856783789557668... . - Vaclav Kotesovec, Dec 18 2014
Showing 1-7 of 7 results.

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

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