A079278 -id:A079278 - cp's OEIS frontend

A080582 Ratios of successive terms of A079278.

2, 5, 31, 1171, 1638151, 3146427633211, 11417451157987217266902031, 148220146017562180410722652131732909019133677172771, 24679246312469754122885261982560498845861213427729059585299568584375038492624323416323464976525500951

Offset: 1

N. J. A. Sloane, Feb 22 2003

nonn

A079269 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(1+1/b(n-1)); sequence gives numerator of b(n).

Original entry on oeis.org

1, 3, 21, 861, 1275141, 2551762438701, 9546380157472159016030421, 126857284256055227389078067834858327568823447932861

Offset: 1

N. J. A. Sloane, Feb 16 2003

The next term is too large to include.

			The b sequence begins 1, 3/2, 21/10, 861/310, 1275141/363010, 2551762438701/594665194510, ... = a(n)/A079278.

Leroy Quet, a(m) =a(m-1)+1/(1+1/a(m-1)), posting to SeqFan mailing list, Feb 15 2003.

Cf. A079278, A080581, A080582, A080984, A080985.

b := proc(n) option remember; if n=1 then 1 else b(n-1)+1/(1+1/b(n-1)); fi; end;

nxt[n_]:=n+1/(1+1/n); Numerator/@Nest[Append[#,nxt[Last[#]]]&,{1},10]  (* Harvey P. Dale, Apr 21 2011 *)

Conjecture: a(m+1) = a(m)^2 + a(m)^3 /(2a(m-1)^2) - a(m)a(m-1)^2/2 for m >= 2. - Leroy Quet

A080984 Define b by b(1) = 1 and for n > 1, b(n) = b(n-1) + 1/(2 + 1/b(n-1)); sequence gives numerator of b(n).

Original entry on oeis.org

1, 4, 56, 9968, 294115808, 242590126064151488, 158248601344912132157178428071499648, 65129411362626329768830076910903417752818896343320137665280356705971968

Offset: 1

Hugo Pfoertner, Feb 26 2003

The next term has 285 digits. - Harvey P. Dale, Jul 07 2011

			The sequence {b(n)} begins 1, 4/3, 56/33, 9968/4785, 294115808/118289985, ...

Leroy Quet, Sequence & Its Numerators & Denominators, posting to SeqFan mailing list, Feb 26 2003.

Cf. A080985, A080986, A080987, A079269, A079278.

Numerator/@NestList[#+1/(2+1/#)&,1,9] (* Harvey P. Dale, Jul 07 2011 *)

a := 1; for i := 1:8 do write a := a+1/(2+1/a);

b(k) = n(k)/d(k); n(1)=1, d(1)=1, m=2; for k >= 2: n(k+1) = n(k) *(m*n(k) + 2*d(k)), d(k+1) = d(k)*(m*n(k) + d(k)). - Leroy Quet

A080985 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(2+1/b(n-1)); sequence gives denominator of b(n).

Original entry on oeis.org

1, 3, 33, 4785, 118289985, 83574429584465985, 47533348097639173195113266868291585, 17303590888755815609063515462865866934559547198082397287546077959536385

Offset: 1

Hugo Pfoertner, Feb 26 2003

Suggested by Leroy Quet, Feb 26 2003.

			See under A080984.

Cf. A080984, A080986, A080987, A079269, A079278.

See under A080984.

A080988 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(3+1/b(n-1)); sequence gives numerator of b(n).

Original entry on oeis.org

1, 5, 115, 57155, 13457544835, 718532108172999980195, 1987460976488531436231264449305834729789315, 14835338180729281137836887250133924105479472089418750626398379615457041439472496214755

Offset: 1

Hugo Pfoertner, Feb 26 2003

Suggested by Leroy Quet, Feb 26 2003.

			The sequence begins 1, 5/4, 115/76, 57155/31996, 13457544835/6509938156, ...

Cf. A080989, A080990, A080991, A079269, A079278.

NestList[#+1/(3+1/#)&,1,10]//Numerator (* Harvey P. Dale, Dec 13 2018 *)

a := 1; for i := 1:8 do write a := a+1/(3+1/a);

b[k]=n[k]/d[k]; n[1]=1, d[1]=1, m=3; for k>=2: n[k+1] = n[k] *(m*n[k] + 2*d[k]), d[k+1] = d[k] *(m*n[k] + d[k])

A080989 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(3+1/b(n-1)); sequence gives denominator of b(n).

Original entry on oeis.org

1, 4, 76, 31996, 6509938156, 305202648617286353116, 751042364315893065602546739334386372522316, 5042066805299767273773988103135929366654804375467223444639109745553463308647365564476

Offset: 1

Hugo Pfoertner, Feb 26 2003

Suggested by Leroy Quet, Feb 26 2003.

			See under A080988.

Cf. A080988, A080990, A080991, A079269, A079278.

NestList[#+1/(3+1/#)&,1,10]//Denominator (* Harvey P. Dale, Apr 19 2023 *)

See under A080988.

A355615 Define a rational sequence {b(n)} as b(1) = 1, b(n) = b(n-1) + 1/(n + 1 - b(n-1)) for n > 1; a(n) is the numerator of b(n).

Original entry on oeis.org

1, 3, 19, 689, 902919, 1610893922869, 5422187846648306990942459, 65408471597507349805723190837012905483968615226329

Offset: 1

Leonid Broukhis, Jul 09 2022

The next term is too large to include.

			The sequence {b(n)} begins 1, 3/2, 19/10, 689/310, 902919/363010, 1610893922869/594665194510, ...

Cf. A079278 (denominators).

cp's OEIS Frontend

A080582 Ratios of successive terms of A079278.

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A079269 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(1+1/b(n-1)); sequence gives numerator of b(n).

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A080984 Define b by b(1) = 1 and for n > 1, b(n) = b(n-1) + 1/(2 + 1/b(n-1)); sequence gives numerator of b(n).

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A080985 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(2+1/b(n-1)); sequence gives denominator of b(n).

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A080988 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(3+1/b(n-1)); sequence gives numerator of b(n).

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A080989 Define b by b(1) = 1 and for n>1, b(n) = b(n-1)+1/(3+1/b(n-1)); sequence gives denominator of b(n).

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A355615 Define a rational sequence {b(n)} as b(1) = 1, b(n) = b(n-1) + 1/(n + 1 - b(n-1)) for n > 1; a(n) is the numerator of b(n).

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