A003095 -id:A003095 - cp's OEIS frontend

A329429 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 2, 2, 1, 5, 8, 8, 4, 1, 26, 80, 144, 168, 138, 80, 32, 8, 1, 677, 4160, 13888, 31776, 54792, 74624, 82432, 74944, 56472, 35296, 18208, 7664, 2580, 672, 128, 16, 1, 458330, 5632640, 36109952, 158572864, 531441232, 1439520512, 3264101376, 6342205824

Offset: 0

Clark Kimberling, Nov 13 2019

Let f(x) = x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Except for the first term, the sequence (p(n,0)) = (1, 1, 5, 26, 677, ...) is found in A003095 and A008318. This is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

			Rows 0..4:
   1;
   1,  1;
   2,  2,   1;
   5,  8,   8,   4,   1;
  26, 80, 144, 168, 138, 80, 32, 8, 1.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + x^2,
  2 + 2 x^2 + x^4,
  5 + 8 x^2 + 8 x^4 + 4 x^6 + x^8,
  26 + 80 x^2 + 144 x^4 + 168 x^6 + 138 x^8 + 80 x^10 + 32 x^12 + 8 x^14 + x^16.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Cf. A003095, A008318, A329430, A329431, A329432, A329433.

f[x_] := x^2 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329429 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 7}]  (* A329429 array *)

p(n,0)  = (1, 1, 2, 5, 26, 677, 458330, ...)
p(n,1)  =    (1, 2, 5, 26, 677, 458330, ...)
p(n,2)  =       (2, 5, 26, 677, 458330, ...)
p(n,5)  =          (5, 26, 677, 458330, ...)
p(n,26) =             (26, 677, 458330, ...), etc.;
that is, p(n,p(k,0)) = p(n+k-2,0); there are similar identities for other sequences p(n,h).

A137560 Let f(z) = z^2 + c, then row k lists the expansion of the n-fold composition f(f(...f(0)...)) in rising powers of c.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 6, 6, 4, 1, 0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1, 0, 1, 1, 2, 5, 14, 42, 100, 221, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788

Offset: 0

Roger L. Bagula, Apr 25 2008

The root of one of these polynomials gives Julia Douady's rabbit.
These polynomials are basic to the theory of "cycles" in complex dynamics.
These polynomials are also described in a comment by Donald D. Cross in the entry for the Catalan numbers, A000108.
Except for the first row, row sums are A003095 (a(n) = a(n-1)^2 + 1). - Gerald McGarvey, Sep 26 2008
The coefficients also enumerate the ways to divide a line segment into at most j pieces, with 0 <= j <= 2^n, in which every piece is a power of two in size (for example, 1/4 is allowed but 3/8 is not), no piece is less than 1/2^n of the whole, and every piece is aligned on a power of 2 boundary (so 1/4+1/2+1/4=1 is not allowed). See the everything2 web link (which treats the segment as a musical measure). - Robert Munafo, Oct 29 2009
Also the number of binary trees with exactly J leaf nodes and a height no greater than N. See the Munafo web page and note the connection to A003095. - Robert Munafo, Nov 03 2009
The sequence of polynomials is conjectured to tend to the Catalan numbers (A000108). - Jon Perry, Oct 31 2010
It can be shown that the initial n nonzero terms of row n are the first Catalan numbers. - Joerg Arndt, Jun 04 2016
From Jianing Song, Mar 23 2021: (Start)
Let P_0(z) = 0, P_{n+1}(z) = P_n(z)^2 + z for n >= 0. For n > 0, the n-th row gives the coefficients of P_n(z) (a polynomial with degree 2^(n-1) for n > 0) in rising powers of z. Note that the famous Mandelbrot set is Intersect_{n>=0} {z: |P_n(z)| <= 2}. In particular, the Mandelbrot set is compact since it is closed and bounded.
Let P(z) = (1 - sqrt(1-4*z))/2. For every 0 < r < 1/4, P_n(z) converges uniformly to P(z) on the disk {z: |z| <= r}, because |P_n(z) - P(z)| <= (1/2)*(1 - sqrt(1-4*r))^(n+1) for every |z| <= r. Note that P(z)/z is the generating function for Catalan numbers, which explains the comment from Joerg Arndt above. Is the convergence uniform on the disk {z: |z| <= 1/4}? (End)

			Triangle starts:
  {1},
  {0, 1},
  {0, 1, 1},
  {0, 1, 1, 2, 1},
  {0, 1, 1, 2, 5, 6, 6, 4, 1},
  {0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
  {0, 1, 1, 2, 5, 14, 42, 100, 221, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1},
  ...

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, pp 128-129

Alois P. Heinz, Rows n = 0..13, flattened (rows n=0..8 from Roger L. Bagula)
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Trans., Vol. 1, No. 1, Article 1 (July 2021).
Robert Munafo, Lemniscates [From _Robert Munafo_, Oct 29 2009]
Everything2 user ferrouslepidoptera, How many melodies are there in the universe? [From _Robert Munafo_, Oct 29 2009]
Wikipedia, Mandelbrot set

A052154 gives the same array read by antidiagonals.
A137867 gives the related Misiurewicz polynomials. [From Robert Munafo, Dec 12 2009]
Cf. A202019 (reversed rows).
Cf. A309049.

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
      x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= n-> `if`(n=0, 1, (m-> (p-> seq(coeff(p, x, m-i),
                  i=-1..m))(b(m)))(2^(n-1)-1)):
seq(T(n), n=0..7);  # Alois P. Heinz, Jul 11 2019

f[z_] = z^2 + x; g = Join[{1}, ExpandAll[NestList[f, x, 7]]]; a = Table[CoefficientList[g[[n]], x], {n, 1, Length[g]}]; Flatten[a] Table[Apply[Plus, CoefficientList[g[[n]], x]], {n, 1, Length[g]}];

p = vector(6); p[1] = x; for(n=2,6, p[n] = p[n-1]^2 + x); print1("1"); for(n=1,6, for(m=0,poldegree(p[n]), print1(", ",polcoeff(p[n],m)))) \\ Gerald McGarvey, Sep 26 2008

Edited by N. J. A. Sloane, Apr 26 2008

Offset set to 0 and new name from Joerg Arndt, Jun 04 2016

A056207 Number of binary trees of height <= n.

Original entry on oeis.org

3, 24, 675, 458328, 210066388899, 44127887745906175987800, 1947270476915296449559703445493848930452791203, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352024

Offset: 1

Todd K. Moon (Todd.Moon(AT)ece.usu.edu), Aug 02 2000

Todd K. Moon, "Enumerations of binary trees, types of trees and the number of reversible variable length codes," submitted to Discrete Applied Mathematics, 2000.

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001699, A002449, A003095.

from itertools import accumulate
def f(anm1, _): return anm1**2 + 4*anm1 + 3
def aupton(terms): return list(accumulate([3]*terms, f))
print(aupton(8)) # Michael S. Branicky, Mar 24 2021

a(n) = d(n) + a(n-1), d(n) = A001699(n) is the number of binary trees of depth exactly n.

a(n) = A003095(n+2) - 2 = A004019(n+1) - 1 = a(n-1)^2 + 4*a(n-1) + 3.

More terms from Henry Bottomley, Jul 09 2001

A098152 a(n) = a(n-1)^2 + n, with a(0)=0.

Original entry on oeis.org

0, 1, 3, 12, 148, 21909, 480004287, 230404115538378376, 53086056457022411804685755744397384, 2818129390158170901506703075470572449397357853477615482257305306043465

Offset: 0

Henry Bottomley, Oct 25 2004

nonn

			a(4) = a(3)^2 + 4 =12^2 + 4 = 148.

Indranil Ghosh, Table of n, a(n) for n = 0..12
Olivier Bodini, Danièle Gardy, Bernhard Gittenberger, Zbigniew Gołębiewski, On the number of unary-binary tree-like structures with restrictions on the unary height, arXiv:1510.01167 [math.CO], 2015, see Table 1, p. 19.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A153058, A153059, A153060, A086851.

[0] cat [n eq 1 select 1 else Self(n-1)^2+n: n in [1..10]]; // Vincenzo Librandi, Oct 06 2015

a=0; lst={}; Do[a=a^2+n; AppendTo[lst, a], {n, 0, 11}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
RecurrenceTable[{a[0]==0,a[n]==a[n-1]^2+n},a,{n,10}] (* Harvey P. Dale, Jul 28 2012 *)

For n>0, a(n) = floor(1.366609561487624975914833969579996...^(2^n)) = floor(A028300(n)^0.68178667449368682115305109818...) = ceiling(A003095(n)^1.53346965582393874689368175542252...).

A101189 G.f. A(x) is defined as the limit A(x) = lim_{n->oo} F(n)^(1/2^(n-1)) where F(n) is defined by F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1.

Original entry on oeis.org

1, 2, 0, 4, -8, 16, -40, 144, -512, 1696, -5696, 19840, -70048, 247744, -880128, 3152768, -11386624, 41389568, -151273728, 555794944, -2052141056, 7610274816, -28331018240, 105833345024, -396594444800, 1490425179136, -5615651143680, 21209004267520, -80276663808000

Offset: 0

Paul D. Hanna, Dec 03 2004

sign

Sequences A101190 and A101191 are related to doubly exponential numbers A003095 and to Catalan numbers (A000108).

			G.f.: A(x) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...
GENERATING METHOD.
We can illustrate the generating method for g.f. A(x) as follows.
Given F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1,
the first few polynomials generated by F(n) begin
F(0) = 1,
F(1) = F(0)^2 + (2*x)^(2^1-1) = 1 + 2*x,
F(2) = F(1)^2 + (2*x)^(2^2-1) = 1 + 4*x + 4*x^2 + 8*x^3,
F(3) = F(2)^2 + (2*x)^(2^3-1) = 1 + 8*x + 24*x^2 + 48*x^3 + 80*x^4 + 64*x^5 + 64*x^6 + 128*x^7,
F(4) = F(3)^2 + (2*x)^(2^4-1) = = 1 + 16*x + 112*x^2 + 480*x^3 + 1504*x^4 + 3712*x^5 + 7296*x^6 + 12032*x^7 + 17664*x^8 + 22528*x^9 + 26624*x^10 + 28672*x^11 + 20480*x^12 + 16384*x^13 + 16384*x^14 + 32768*x^15,
...
and the 2^(n-1)-th root of F(n) yields the series shown by
F(1)^(1/2^0) = 1 + 2*x,
F(2)^(1/2^1) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 112*x^7 - 320*x^8 + 928*x^9 - 2752*x^10 + 8320*x^11 - 25504*x^12 + ...,
F(3)^(1/2^2) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...,
F(4)^(1/2^3) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...,
...
The limit of this process tends to the g.f. A(x).

Paul D. Hanna, Table of n, a(n) for n = 0..1025

Cf. A101190, A101191, A005187, A004134, A003095.

{a(n) = my(F=1,A,L); if(n==0,A=1, L = ceil(log(n+1)/log(2)); for(k=1,L, F = F^2 + (2*x)^(2^k-1)  +x*O(x^n)); A = polcoeff(F^(1/(2^(L-1))),n)); A}
for(n=0,32, print1(a(n),", "))

G.f. A(x) = ( Sum_{n>=0} A101190(n)/2^A005187(n) * (2*x)^n )^2.

G.f. A(x) = ( Sum_{n>=0} A101191(n)/2^A004134(n) * (2*x)^n )^4.

Entry revised by Paul D. Hanna, Mar 05 2024

A158985 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, 2, 1, 4, 8, 8, 5, 1, 8, 32, 80, 138, 168, 144, 80, 26, 1, 16, 128, 672, 2580, 7664, 18208, 35296, 56472, 74944, 82432, 74624, 54792, 31776, 13888, 4160, 677, 1, 32, 512, 5440, 43048, 269920, 1393728, 6082752, 22860480, 75010560, 217147904

Offset: 1

Clark Kimberling, Apr 02 2009

			Row 1: 1 1 (from x + 1)
Row 2: 1 2 2 (from x^2 + 2*x + 2)
Row 3: 1 4 8 8 5
Row 4: 1 8 32 80 138 168 144 80 26

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

Cf. A158982, A158983, A158984, A158986, A003095 (row sums), A062013.

tabf(nn) = {my(P = x+1); print(Vec(P)); for (n=1, nn, P = 1 + P^2; print(Vec(P)););} \\ Michel Marcus, Jul 01 2015

From Peter Bala, Jul 01 2015: (Start)
P(n,x) = P(n,-2 - x) for n >= 2.
P(n+1,x)= P(n,(1 + x)^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.
n-th row sum = P(n,1) = A003095(n+1);
P(n,1) = P(n+1,0) = P(n+1,-2); P(n,1) = P(n,-3) for n >= 2.
P(n,2) = A062013(n). (End)

A213437 Nonlinear recurrence: a(n) = a(n-1) + (a(n-1)+1)*Product_{j=1..n-2} a(j).

Original entry on oeis.org

1, 3, 7, 31, 703, 459007, 210066847231, 44127887746116242376703, 1947270476915296449559747573381594836628779007

Offset: 1

N. J. A. Sloane, Jun 11 2012

nonn

This sequence was going to be included in the Aho-Sloane paper, but was omitted from the published version.

It appears that the sequence becomes periodic mod 10^k for any k, with period 3. The last digits are (1,3,7) repeated. Modulo 10^5 the sequence enters the cycle (56703, 79007, 23231) after the first 10 terms. - M. F. Hasler, Jul 23 2012. See also A214635, A214636.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437. [Includes many similar sequences, although not this one.]
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!)

Cf. A076949, A214635, A214636, A003095, A001699.

A213437 := proc(n)
        if n = 1 then 1;
        else procname(n-1)+(1+procname(n-1))*mul(procname(j),j=1..n-2);
        end if;
end proc: # R. J. Mathar, Jul 23 2012

RecurrenceTable[{a[n] == a[n-1]+(a[n-1]+1)*(a[n-1]-a[n-2])*a[n-2]/(a[n-2]+1),a[1]==1,a[2]==3},a,{n,1,10}] (* Vaclav Kotesovec, May 06 2015 *)

a=[1];for(n=1,11,a=concat(a, a[n] + (a[n]+1) * prod(k=1,n-1, a[k] )));a \\ - M. F. Hasler, Jul 23 2012

a(n) = a(n-1)+(a(n-1)+1)*(a(n-1)-a(n-2))*a(n-2)/(a(n-2)+1). - Johan de Ruiter, Jul 23 2012
a(2+3k) = 9007 (mod 10^4) for all k>0. - M. F. Hasler, Jul 23 2012
a(n) ~ c^(2^n), where c = A076949 = 1.2259024435287485386279474959130085213212293209696612823177009... . - Vaclav Kotesovec, May 06 2015
a(n) = A001699(n)/A001699(n-1); a(n+1) - a(n) = A001699(n) + A001699(n-1); a(n) = A003095(n) + A003095(n-1). - Peter Bala, Feb 03 2017

Definition recovered by Johan de Ruiter, Jul 23 2012

A008318 Smallest number strictly greater than previous one which is the sum of squares of two previous distinct terms (a(1)=1, a(2)=2).

Original entry on oeis.org

1, 2, 5, 26, 29, 677, 680, 701, 842, 845, 866, 1517, 458330, 458333, 458354, 459005, 459170, 462401, 462404, 462425, 463076, 463241, 491402, 491405, 491426, 492077, 492242, 708965, 708968, 708989, 709640, 709805, 714026, 714029, 714050, 714701

Offset: 1

R. Muller

A003095 is a subsequence apart from the initial term. - Reinhard Zumkeller, Jan 17 2008

The subsequence of primes begins: 2, 5, 29, 677, 701, 458333, 462401, 492077, 708989, 714029, ... - Jonathan Vos Post, Nov 21 2012

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

T. D. Noe, Table of n, a(n) for n=1..1000
Mihaly Bencze [Beneze], Smarandache Recurrence Type Sequences, in Bull. Pure Appl. Sciences, Vol. 16E, No. 2, 231-236, 1997.
F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ...
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences.

Cf. A192476.

import Data.Set (singleton, deleteFindMin, insert)
a008318 n = a008318_list !! (n-1)
a008318_list = f [1] (singleton 1) where
   f xs s =
     m : f (m:xs) (foldl (flip insert) s' (map (+ m^2) (map (^ 2) xs)))
     where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

a[1]=1; a[2]=2; a[n_] := a[n] = First[ Select[ Sort[ Flatten[ Table[a[j]^2 + a[k]^2, {j, 1, n-1}, {k, j+1, n-1}]]], # > a[n-1] & , 1]]; Table[a[n], {n, 1, 36}](* Jean-François Alcover, Nov 10 2011 *)

More terms from David W. Wilson

A077496 Decimal expansion of lim_{n -> infinity} A001699(n)^(1/2^n).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Dec 01 2002

			1.5028368010497564997529364237321694087388717439635793069906714243...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, p. 443.

G. C. Greubel, Table of n, a(n) for n = 1..10000
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!)
Anna de Mier and Marc Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, El. Notes Discr. Math., Vol. 34 (2009), pp. 489-493, Proposition 2.2.
Eric Weisstein's World of Mathematics, Quadratic Recurrence Equation.

Cf. A001699, A003095, A076949.

function A003095(n)
  if n eq 0 then return 0;
  else return 1 + A003095(n-1)^2;
  end if; return A003095;
end function;
function S(n)
  if n eq 1 then return Log(2)/2;
  else return S(n-1) + Log(1 + 1/A003095(n)^2)/2^n;
  end if; return S;
end function;
SetDefaultRealField(RealField(120)); Exp(S(12)); // G. C. Greubel, Nov 29 2022

digits = 105; Clear[b, beta]; b[0] = 1; b[n_] := b[n] = b[n-1]^2 + 1; b[10]; beta[n_] := beta[n] = b[n]^(2^(-n)); beta[5]; beta[n = 6]; While[ RealDigits[beta[n], 10, digits+5] != RealDigits[beta[n-1], 10, digits+5], Print["n = ", n]; n = n+1]; RealDigits[beta[n], 10, digits] // First (* Jean-François Alcover, Jun 18 2014 *)
(* Second program *)
A003095[n_]:= A003095[n]= If[n==0, 0, 1 + A003095[n-1]^2];
S[n_]:= S[n]= If[n==1, Log[2]/2, S[n-1] + Log[1 + 1/A003095[n]^2]/2^n];
RealDigits[Exp[S[13]], 10, 120][[1]] (* G. C. Greubel, Nov 29 2022 *)

@CachedFunction
def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
@CachedFunction
def S(n): return log(2)/2 if (n==1) else S(n-1) + log(1 + 1/(A003095(n))^2)/2^n
numerical_approx( exp(S(12)), digits=120) # G. C. Greubel, Nov 29 2022

Equals A076949^2. - Vaclav Kotesovec, Dec 17 2014

Equals exp(Sum_{k>=1} log(1+1/A003095(k)^2)/2^k) (Aho and Sloane, 1973). - Amiram Eldar, Feb 02 2022

A062013 a(1) = 3, a(n) = a(n-1)^2 + 1.

Original entry on oeis.org

3, 10, 101, 10202, 104080805, 10832813969448026, 117349858496668297583751115296677, 13770989289188072635789270799554901596999795308252284177727242330

Offset: 1

Amarnath Murthy, Jun 01 2001

nonn

Define f(n, c) := x - Sum_{k>=0} A088674(k)/(2*x)^(2*k+1) where x = c^(2^n). Then a(n) = f(n, 1.78050350...). - Michael Somos, Jun 07 2021

			a(4) = a(3)^2 +1 = 101^2 +1 =10202

Harry J. Smith, Table of n, a(n) for n = 1..11

Cf. A003095, A088674.

NestList[#^2+1&,3,10] (* Harvey P. Dale, Feb 21 2013 *)
a[ n_] := If[n < 2, 3 Boole[n == 1], A062013[n - 1]^2 + 1]; (* Michael Somos, Jun 07 2021 *)

{ for (n=1, 11, if (n==1, a=3, a=a^2 + 1); write("b062013.txt", n, " ", a) ) } [Harry J. Smith, Jul 29 2009]

a(n) ~ c^(2^n), where c = 1.78050350352842911667602268320603615359... - Vaclav Kotesovec, Sep 20 2013

More terms from Jason Earls, Jun 02 2001
Offset changed from 0,1 to 1,1 by Harry J. Smith, Jul 29 2009
Typo in a(8) corrected by N. J. A. Sloane, Aug 31 2009 using the b-file.

cp's OEIS Frontend

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