A005605 -id:A005605 - cp's OEIS frontend

A058182 a(n) = a(n-1)^2 + a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.

1, 0, 1, 1, 2, 5, 27, 734, 538783, 290287121823, 84266613096281243382112, 7100862082718357559748563880517486086728702367, 50422242317787290639189291009890702507917377925161079229314384058371278254659634544914784801

Offset: 0

Henry Bottomley, Nov 15 2000

Has property that CONTINUANT([1, 1, 2, 5, 27, 734, 538783, ...]) = [1, 2, 5, 27, 734, 538783, ...]. - N. J. A. Sloane Jul 19 2002

For n > 2, a(n) is the numerator of the simplified continued fraction resulting from [a(2), a(3), ..., a(n)]. Therefore, for n > 2, a(n) represents the number of ways to tile a (n-2)-board with dominoes and stackable squares, where nothing can be stacked on a domino but otherwise for 2 < i < n, the i-th cell may be stacked by as many as a(i) squares (see Benjamin, A. and Quinn, J.). - Melvin Peralta, Feb 22 2016

			a(6) = a(5)^2 + a(4) = 5^2 + 2 = 27.

Arthur Benjamin and Jennifer Quinn, Proofs that Really Count, Mathematical Association of America, 2003, see pages 49-51.

Melvin Peralta, Table of n, a(n) for n = 0..15
N. J. A. Sloane, Transforms
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000278, A005605, A058181.

I:=[1,0]; [n le 2 select I[n] else Self(n-1)^2+Self(n-2): n in [1..13]]; // Vincenzo Librandi, Feb 23 2016

Join[{a=1,b=0},Table[c=a+b^2;a=b;b=c,{n,12}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
Join[{1},Transpose[NestList[{Last[#],Last[#]^2+First[#]}&,{0,1},12]][[1]]] (* Harvey P. Dale, May 15 2011 *)
RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == a[n-1]^2 + a[n-2]}, a, {n, 13}] (* Vincenzo Librandi, Feb 23 2016 *)

a(n)=if(n<0, -a(-1-n), if(n<2, 1-n, a(n-1)^2+a(n-2))) /* Michael Somos, May 05 2005 */

a(n)^2 = a(n+1) - a(n-1), a(-1-n) = -a(n).
For n > 1, a(n+1) = floor(c^(2^n)) where c=1.108604586393628626769904017539.... - Benoit Cloitre, Nov 30 2002
a(n+1) = a(n)^2 + floor(sqrt(a(n))) = A000290(a(n)) + A000196(a(n)) for n > 2. - Reinhard Zumkeller, May 16 2006

More terms from Reinhard Zumkeller, May 16 2006

A243139 a(n) = 2^prime(n) + prime(n).

Original entry on oeis.org

6, 11, 37, 135, 2059, 8205, 131089, 524307, 8388631, 536870941, 2147483679, 137438953509, 2199023255593, 8796093022251, 140737488355375, 9007199254741045, 576460752303423547, 2305843009213694013, 147573952589676412995, 2361183241434822606919

Offset: 1

Vincenzo Librandi, Jun 03 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
J. Mestel, Archimedeans Problems Drive 1977, Eureka, 39 (1978), 38-40. (Annotated scanned copy)

Cf. A034785, A100105.

Magma
```
[2^p + p: p in PrimesUpTo(80)];
```

f[n_]:=(2^Prime[n] + Prime[n]); Array[f, 80, 1]

A259984 a(n) = A000787(n) + 1.

Original entry on oeis.org

1, 2, 9, 12, 70, 89, 97, 102, 112, 182, 610, 620, 690, 809, 819, 889, 907, 917, 987, 1002, 1112, 1692, 1882, 1962, 6010, 6120, 6700, 6890, 6970, 8009, 8119, 8699, 8889, 8969, 9007, 9117, 9697, 9887, 9967, 10002, 10102, 10802, 11012, 11112, 11812, 16092

Offset: 1

N. J. A. Sloane, Jul 13 2015

J. Mestel, Archimedeans Problems Drive 1977, Eureka, 39 (1978), 38-40. (Annotated scanned copy)

Cf. A000787.

cp's OEIS Frontend

A058182 a(n) = a(n-1)^2 + a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A243139 a(n) = 2^prime(n) + prime(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A259984 a(n) = A000787(n) + 1.

Views

Author

Keywords

Links

Crossrefs