A003096 -id:A003096 - cp's OEIS frontend

A077124 Decimal expansion of the constant c such that A003096(n-1) = ceiling(c^(2^n)).

1, 2, 9, 5, 5, 5, 3, 5, 3, 6, 1, 8, 6, 5, 3, 2, 5, 4, 1, 3, 9, 8, 1, 5, 5, 9, 7, 0, 0, 5, 9, 3, 3, 5, 3, 9, 4, 7, 7, 6, 8, 6, 5, 3, 6, 0, 0, 1, 5, 4, 4, 6, 2, 6, 4, 9, 8, 1, 8, 7, 0, 1, 0, 3, 4, 9, 7, 3, 6, 8, 9, 3, 9, 5, 4, 5, 4, 0, 5, 6, 4, 6, 4, 5, 4, 9, 3, 9, 1, 3, 4, 1, 1, 1, 2, 3, 7, 0, 7, 2, 1, 2, 4, 4, 9

Offset: 1

Benoit Cloitre, Nov 29 2002

Charles R Greathouse IV, Table of n, a(n) for n = 1..300001.

Cf. A003096.

c = 1.2955535361865325413981559700593353....;

c = lim_{n -> infinity} A003096(n-1)^(1/2^n).

A158984 Coefficients of polynomials (in descending powers of x) P(n,x) := -1 + P(n-1,x)^2, where P(1,x) = x - 1.

Original entry on oeis.org

1, -1, 1, -2, 0, 1, -4, 4, -1, 1, -8, 24, -32, 14, 8, -8, 0, 0, 1, -16, 112, -448, 1116, -1744, 1552, -384, -700, 736, -160, -128, 64, 0, 0, 0, -1, 1, -32, 480, -4480, 29112, -139552, 509600, -1441024, 3166616, -5345344, 6668992, -5473536, 1494624

Offset: 1

Clark Kimberling, Apr 02 2009

			Row 1: 1 -1 (from x-1)
Row 2: 1 -2 0 (from x^2-2x)
Row 3: 1 -4 4 -1
Row 4: 1 -8 24 -32 14 8 -8 0 0

Clark Kimberling, Polynomials defined by a second-order recurrence, interlacing zeros, and Gray codes, The Fibonacci Quarterly 48 (2010) 209-218.

Cf. A158982, A158983, A158985, A158986, A003096.

tabf(nn) = {p = x-1; print(Vec(p)); for (n=2, nn, p = -1 + p^2; print(Vec(p)););} \\ Michel Marcus, Mar 01 2016

From Peter Bala, Jul 01 2015: (Start)
P(n,x) = P(n,2 - x) for n >= 2.
P(n+1,x)= P(n,(x - 1)^2). Thus if alpha is a zero of P(n,x) then sqrt(alpha) + 1 is a zero of P(n+1,x).
Define a sequence of polynomials Q(n,x) by setting Q(1,x) = -1 + x^2 and Q(n,x) = Q(n-1, -1 + x^2) for n >= 2. Then P(n,x) = Q(n,sqrt(x)).
Q(n,x) = Q(k,Q(n-k,x)) for 1 <= k <= n - 1; P(n,x) = P(k,P(n-k,x)^2) for 1 <= k <= n - 1.
P(n,x)^(2^k) divides P(n + 2*k,x) in Z[x] for k = 1,2,....
P(n,4) = A003096(n). (End)

A139244 a(0) = 4; a(n) = a(n-1)^2 - 1.

Original entry on oeis.org

4, 15, 224, 50175, 2517530624, 6337960442777829375, 40169742574216538983356186036612890624, 1613608218478824775913354216413699241125577233045500390244103887844987109375

Offset: 0

Jonathan Vos Post, Jun 06 2008

This is the next analog of A003096 with different initial value a(0), as starting with a(0) = 2 is A003096 and a(0) = 3 is A003096 with first term omitted. It alternates between even and odd values, specifically between 4 mod 10 and 5 mod 10 and is always composite (by difference of squares factorization).

a(n+2) is divisible by a(n)^2. A007814(a(2 n)) = A153893(n). - Robert Israel, Jul 20 2015

Robert Israel, Table of n, a(n) for n = 0..10
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A003096, A007814, A153893, A260315.

A[0]:= 4:
for n from 1 to 10 do A[n]:= A[n-1]^2-1 od:
seq(A[i],i=0..10); # Robert Israel, Jul 20 2015

a=4; lst={a}; Do[b=a^2-1; AppendTo[lst,b]; a=b, {n,10}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)

a(n)=if(n,a(n-1)^2-1,4) \\ Charles R Greathouse IV, Jul 23 2015

a(n-1) = ceiling(c^(2^n)) where c is a constant between 1 and 2.

More specifically, c=1.9668917617901763653335057202... (sequence A260315). - Chayim Lowen, Jul 17 2015

A186750 a(0) = 3; thereafter, a(n) = a(n-1)^2 - 3.

Original entry on oeis.org

3, 6, 33, 1086, 1179393, 1390967848446, 1934791555410494424614913, 3743418362887760317407541271559358491868341997566

Offset: 0

Jonathan Vos Post, Feb 26 2011

This is to A001566 as 3 is to 2 (subtrahend). Unlike A001566, which begins with 4 consecutive primes, this sequence can never be prime after a(0) = 3, because the first two terms are both multiples of 3, hence all later terms are. This is the k = 3 row of the array A(k, 0) = 3, A(k, n) = A(k, n-1)^2 - k; and A001566 is the k = 2 row. A003096(n+1) is the k = 1 row.

Vincenzo Librandi, Table of n, a(n) for n = 0..11
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001566, A003096.

RecurrenceTable[{a[0] == 3, a[n] == a[n-1]^2 - 3}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
Drop[Abs[NestList[#^2 - 3 &, 0, 9]], 1] (* Alonso del Arte, Apr 08 2016 *)

a(n) ~ c^(2^n), where c = 2.3959550115176494685408322564302422183669584045032057908382914927198090627... - Vaclav Kotesovec, Dec 18 2014

A073015 a(n) is such that 2 = sqrt(1+sqrt(1+sqrt(1+....sqrt(a(n))....))) where there are n sqrt's.

Original entry on oeis.org

3, 4, 9, 64, 3969, 15745024, 247905749270529, 61457260521381894004129398784, 3776994870793005510047522464634252677140721938309041881089, 14265690253996672387291309349232388828298289458234016200317876247121873778287073518355813134107244701354409532063744

Offset: 0

Benoit Cloitre, Aug 03 2002

			2 = sqrt(1+sqrt(1+sqrt(64))) hence a(3)=64.

Berndt and Rankin, "Ramanujan, letters and commentary", p. 275
Bruce Berndt, "Ramanujan's notebook", part II, Springer Verlag, pp. 107-112

Cf. A003096.

a073015 n = a073015_list !! n
a073015_list = iterate (\x -> (x - 1) ^ 2) 3  -- Reinhard Zumkeller, Jul 16 2012

a[0] = 3; a[n_] := a[n] = (a[n-1]-1)^2; Table[ a[n], {n, 0, 9}] (* Jean-François Alcover, Dec 14 2011, after Pari *)
NestList[(#-1)^2&,3,10] (* Harvey P. Dale, Feb 04 2012 *)

PARI
```
a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2)
```

a(n) = A003096(n) + 1.

A186751 a(0) = 3; thereafter, a(n) = a(n-1)^2 - 4.

Original entry on oeis.org

3, 5, 21, 437, 190965, 36467631221, 1329888126870853950837, 1768602429992068534155014726612412013000565

Offset: 0

Jonathan Vos Post, Feb 26 2011

This is to A001566 as 4 is to 2 (subtrahend). This is the k=4 row of the array A[k,0] = 3, A[k,n] = A[k,n-1]^2 - k; A186750 is the k=3 row; and A001566 is the k=2 row. A003096(n+1) is the k=1 row.

			a(1) = a(0)^2 - 4 = 3^2 - 4 = 5, which is, like a(0), a prime.

Cf. A001566, A003096, A186750.

NestList[#^2-4&,3,10]  (* Harvey P. Dale, Mar 14 2011 *)

A135378 Main diagonal of "square and add k" array.

Original entry on oeis.org

2, 5, 38, 2707, 21418388, 3000279372337641, 255122481276683701099886061668842

Offset: 0

Jonathan Vos Post, Dec 09 2007

Array of recurrence "start with 2, square and add k" begins:
k..|.A[k,n]=A[k,n-1]^2 + k
-1.|.2..3...8....63.......3968..15745023.247905749270528.............A003096
.|.2..4..16...256......65536..4294967296.18446744073709551616......A001146
.|.2..5..26...677.....458330..210066388901.44127887745906175987802.A003095
.|.2..6..38..1446....2090918.4371938082726...19113842599189892819591078...
.|.2..7..52..2707....7327852.53697414933907..2883412370584178505178284652.
.|.2..8..68..4628...21418388.458747344518548.210449126102819371741916028308.
.|.2..9..86..7401...54774806.3000279372337641.9001676312074749038996905444886.
.|.2.10.106.11242..126382570.1597255405035792810...
.|.2.11.128.16391..268664888.72180822044052551...
.|.2.12.152.23112..534164552.285331768613360712..
.|.2.13.178.31693.1004446258.1008912285210202573.
|.2.14.206.42446.1801662926.3245989298922881486.

Cf. A001146, A003095, A003096, A062013.

A[k_,0] = 2; A[k_,n_] := A[k,n] = A[k, n-1]^2 + k; a[n_] := A[n, n]; a /@ Range[0, 6] (* Giovanni Resta, Jun 20 2016 *)

a(n) = A[n,n] where A[k,n] = n-th term of recurrence A[k,0] = 2, A[k,n] = A[k,n-1]^2 + k.

Corrected and edited by Giovanni Resta, Jun 20 2016

cp's OEIS Frontend

A077124 Decimal expansion of the constant c such that A003096(n-1) = ceiling(c^(2^n)).

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A158984 Coefficients of polynomials (in descending powers of x) P(n,x) := -1 + P(n-1,x)^2, where P(1,x) = x - 1.

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A139244 a(0) = 4; a(n) = a(n-1)^2 - 1.

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A186750 a(0) = 3; thereafter, a(n) = a(n-1)^2 - 3.

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A073015 a(n) is such that 2 = sqrt(1+sqrt(1+sqrt(1+....sqrt(a(n))....))) where there are n sqrt's.

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A186751 a(0) = 3; thereafter, a(n) = a(n-1)^2 - 4.

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A135378 Main diagonal of "square and add k" array.

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