A000215 -id:A000215 - cp's OEIS frontend

A152585 Generalized Fermat numbers: 12^(2^n) + 1, n >= 0.

13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137

Offset: 0

Cino Hilliard, Dec 08 2008

There appears to be no divisibility rule for this sequence.

13 is the only prime up to 12^(2^15)+1.

			a(0) = 12^1+1 = 13 = 11(1)+2 = 11(empty product)+2.
a(1) = 12^2+1 = 145 = 11(13)+2.
a(2) = 12^4+1 = 20737 = 11(13*145)+2.
a(3) = 12^8+1 = 429981697 = 11(13*145*20737)+2.
a(4) = 12^16+1 = 184884258895036417 = 11(13*145*20737*429981697)+2.
a(5) = 12^32+1 = 34182189187166852111368841966125057 = 11(13*145*20737*429981697*184884258895036417)+2.

Vincenzo Librandi, Table of n, a(n) for n = 0..12
Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
C. K. Caldwell, "Top Twenty" page, Generalized Fermat Divisors (base=12).
Wilfrid Keller, GFN12 factoring status.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A000215 (Fermat numbers: 2^(2^n)+1, n >= 0).

Cf. A059919, A199591, A078303, A078304, A152581, A080176, A199592.

[12^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Table[12^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

g(a,n) = if(a%2,b=2,b=1);for(x=0,n,y=a^(2^x)+b;print1(y","))

def A152585(n): return (1<<2*(m:=1<Chai Wah Wu, Jul 19 2022

a(0) = 13; a(n)=(a(n-1)-1)^2 + 1, n >= 1.
a(n) = 11*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 11*(empty product, i.e., 1)+ 2 = 13 = a(0). This implies that the terms, all odd, are pairwise coprime. - Daniel Forgues, Jun 20 2011
Sum_{n>=0} 2^n/a(n) = 1/11. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 19 2011

A035089 Smallest prime of form 2^n*k + 1.

Original entry on oeis.org

2, 3, 5, 17, 17, 97, 193, 257, 257, 7681, 12289, 12289, 12289, 40961, 65537, 65537, 65537, 786433, 786433, 5767169, 7340033, 23068673, 104857601, 167772161, 167772161, 167772161, 469762049, 2013265921, 3221225473, 3221225473, 3221225473, 75161927681

Offset: 0

Labos Elemer

nonn

a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order 2^n. - Joerg Arndt, Oct 18 2020

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.

Analogous case is A034694. Fermat primes (A019434) are a subset. See also Fermat numbers A000215.

Cf. A007522, A057775, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k 2^n + 1], {n, 1, 50}]; a (* Artur Jasinski *)

a(n)=for(k=1,9e99,if(ispseudoprime(k<Charles R Greathouse IV, Jul 06 2011

a(0) from Joerg Arndt, Jul 06 2011

A000289 A nonlinear recurrence: a(n) = a(n-1)^2 - 3*a(n-1) + 3 (for n>1).

Original entry on oeis.org

1, 4, 7, 31, 871, 756031, 571580604871, 326704387862983487112031, 106735757048926752040856495274871386126283608871, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068031

Offset: 0

N. J. A. Sloane

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004

This is the special case k=3 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..12
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!)
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Seppo Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000058.

Join[{1}, RecurrenceTable[{a[n] == a[n-1]^2 - 3*a[n-1] + 3, a[1] == 4}, a, {n, 1, 9}]] (* Jean-François Alcover, Feb 06 2016 *)

a(n)=if(n<2,max(0,1+3*n),a(n-1)^2-3*a(n-1)+3)

a(n) = A005267(n) + 2 (for n>0).
a(n) = ceiling(c^(2^n)) + 1 where c = A077141. - Benoit Cloitre, Nov 29 2002
For n>0, a(n) = 3 + Product_{i=0..n-1} a(i). - Vladimir Shevelev, Dec 08 2010

A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, 59649589127497217, 5704689200685129054721, 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321

Offset: 0

N. J. A. Sloane, Dec 30 1999

Alternatively, list of prime factors of terms of A001317 in order of their first appearance. - Labos Elemer, Jan 21 2002
From T. D. Noe, Jan 29 2009: (Start)
That these two definitions give the same sequence follows from the fact (stated as a formula in A001317) that A001317(n) is the product of Fermat numbers F(i) according to which bits of n are set.
For instance, for n=41, the binary representation of n is 101001, which has bits 0, 3 and 5 set. A001317(n) = 3311419785987 = 3*257*4294967297 = F(0)*F(3)*F(5).
This factorization also explains why the "first 31 numbers give odd-sided constructible polygons". I think Hewgill first noticed this factorization. (End)

			Triangle begins:
  3;
  5;
  17;
  257;
  65537;
  641,               6700417;
  274177,            67280421310721;
  59649589127497217, 5704689200685129054721;
  1238926361552897,  93461639715357977769163558199606896584051237541638188580280321;
  ...
A001317(127) = 3*5*17*257*65537.641*6700417*274177*6728042130721, A001317(128) = 59649589127497217*5704689200685129054721. See also A050922. Compare with A053576, where 2 and A000215 appear as prime factors. - _Labos Elemer_, Jan 21 2002

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..29
J. Bernheiden, Fermat Numbers (Text in German)
R. P. Brent, Factorization of the tenth Fermat number
R. P. Brent, Factorization of the eleventh Fermat number
R. P. Brent, Succinct proofs of primality for the factors of some Fermat numbers
R. P. Brent & J. M. Pollard, Factorization of the eighth Fermat number
R. P. Brent et al., Three new factors of Fermat numbers
C. K. Caldwell, The Prime Glossary, Fermat divisor
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012. - From _N. J. A. Sloane_, Jun 13 2012
R. Munafo, Notes on Fermat numbers
Mercedes Orús-Lacort, Fermat numbers are not prime numbers for n >= 5, (2020).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A019434, A023394, A093179.

Cf. A001317, A001316, A003401, A045544, A053576.

Flatten[Transpose[FactorInteger[#]][[1]]&/@Table[2^(2^n)+1,{n,0,8}]] (* Harvey P. Dale, May 18 2012 *)

for(n=0, 1e3, f=factor(2^(2^n)+1)[, 1]; for(i=1, #f, print1(f[i], ", "))) \\ Felix Fröhlich, Aug 16 2014

More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000.
Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of T. D. Noe
Link to Munafo webpage fixed by Robert Munafo, Dec 09 2009

A121270 Prime Sierpinski numbers of the first kind: primes of the form k^k+1.

Original entry on oeis.org

2, 5, 257

Offset: 1

Alexander Adamchuk, Aug 23 2006

Sierpinski proved that k>1 must be of the form 2^(2^j) for k^k+1 to be a prime. All a(n) > 2 must be the Fermat numbers F(m) with m = j+2^j = A006127(j). [Edited by Jeppe Stig Nielsen, Jul 09 2023]

See e.g. pp. 156-157 in M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001. - Walter Nissen, Mar 20 2010

Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Primes of form b*k^k + 1: this sequence (b=1), A216148 (b=2), A301644 (b=3), A301641 (b=4), A301642 (b=16).

Cf. A014566, A048861, A006127, A000215.

Do[f=n^n+1;If[PrimeQ[f],Print[{n,f}]],{n,1,1000}]

for(n=1,9,if(ispseudoprime(t=n^n+1),print1(t", "))) \\ Charles R Greathouse IV, Feb 01 2013

Definition rewritten by Walter Nissen, Mar 20 2010

A090872 a(n) is the smallest number m greater than 1 such that m^(2^k)+1 for k=0,1,...,n are primes.

Original entry on oeis.org

2, 2, 2, 2, 2, 7072833120, 2072005925466, 240164550712338756

Offset: 0

Farideh Firoozbakht, Jan 31 2004

The first five terms of this sequence correspond to Fermat primes.

Note that 7072833120 is not the smallest base to give at least six possibly nonconsecutive k values. For example, 292582836^(2^k) + 1 is prime for k = 0,1,2,3,4,7. - Jeppe Stig Nielsen, Sep 18 2022

			a(5)=7072833120 because 7072833120^2^k+1 for k=0,1,2,3,4,5 are primes.

PrimeGrid, Generalized Fermat Progression Search
Carlos Rivera, Puzzle 137. Product of primes + 1, a square, The Prime Puzzles and Problems Connection.
Carlos Rivera, Prime Puzzle 399. Some more terms, The Prime Puzzles and Problems Connection.

Cf. A019434, A000215.
Cf. A090873, A090874, A090875.
All solutions for fixed n: A006093 (n=0), A070689 (n=1), A070325 (n=2), A070655 (n=3), A070694 (n=4), A235390 (n=5), A335805 (n=6), A337364 (n=7).

a(6) from Jens Kruse Andersen, May 06 2007

a(7) from Kellen Shenton, Aug 13 2020

A000324 A nonlinear recurrence: a(0) = 1, a(1) = 5, a(n) = a(n-1)^2 - 4*a(n-1) + 4 for n>1.

Original entry on oeis.org

1, 5, 9, 49, 2209, 4870849, 23725150497409, 562882766124611619513723649, 316837008400094222150776738483768236006420971486980609

Offset: 0

N. J. A. Sloane

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004
This is the special case k=4 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005
A000058, A000215, A000289 and this sequence here can be represented as values of polynomials defined via P_0(z)= 1+z, P_{n+1}(z) = z+ prod_{i=0..n} P_i(z), with recurrences P_{n+1}(z) = (P_n(z))^2 -z*P_n(z) +z, n>=0. - Vladimir Shevelev, Dec 08 2010

Derek Jennings, Some reciprocal summation identities with applications to the Fibonacci and Lucas numbers, in: G. E. Bergum, Applications of Fibonacci Numbers, Vol. 7, Bergum G. E. et al. (eds.), Kluwer Academic Publishers, 1998, pp. 197-200.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..12
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!)
Daniel Duverney, Irrationality of Fast Converging Series of Rational Numbers, Journal of Mathematical Sciences-University of Tokyo, Vol. 8, No. 2 (2001), pp. 275-316.
Daniel Duverney and Takeshi Kurosawa, Transcendence of infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2002), Article 68.
Solomon W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly, Vol. 70, No. 4 (1963), 403-405.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Seppo Mustonen, On integer sequences with mutual k-residues, 2005.
Seppo Mustonen, On integer sequences with mutual k-residues, 2005. [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ....

a(n) = A001566(n-1)+2 (for n>0).

Cf. A000032, A000058, A000215, A000289.

t = {1, 5}; Do[AppendTo[t, t[[-1]]^2 - 4*t[[-1]] + 4], {n, 11}] (* T. D. Noe, Jun 19 2012 *)
Join[{1}, RecurrenceTable[{a[n] == a[n-1]^2 - 4*a[n-1] + 4, a[1] == 5}, a, {n, 1, 8}]] (* Jean-François Alcover, Feb 07 2016 *)
Join[{1},NestList[#^2-4#+4&,5,10]] (* Harvey P. Dale, Dec 11 2023 *)

a(n)=if(n<2,max(0,1+4*n),a(n-1)^2-4*a(n-1)+4)

a(n)=if(n<1,n==0,n=2^n;fibonacci(n+1)+fibonacci(n-1)+2)

a(n) = L(2^n)+2, if n>0 where L() is Lucas sequence.
For n>=1, a(n) = 4 + Product_{i=0..n-1} a(i). - Vladimir Shevelev, Dec 08 2010
From Amiram Eldar, Sep 10 2022: (Start)
a(n) = Lucas(2^(n-1))^2 for n > 1.
Sum_{n>=1} 4^n/a(n) = 4 (Jennings, 1998; Duverney, 2001). (End)
Product_{n>=1} (1 - 3/a(n)) = 1/4 (Duverney and Kurosawa, 2022). - Amiram Eldar, Jan 07 2023

A152581 Generalized Fermat numbers: a(n) = 8^(2^n) + 1, n >= 0.

Original entry on oeis.org

9, 65, 4097, 16777217, 281474976710657, 79228162514264337593543950337, 6277101735386680763835789423207666416102355444464034512897

Offset: 0

Cino Hilliard, Dec 08 2008

These numbers are all composite. We rewrite 8^(2^n) + 1 = (2^(2^n))^3 + 1.

Then by the identity a^n + b^n = (a+b)*(a^(n-1) - a^(n-2)*b + ... + b^(n-1)) for odd n, 2^(2^n) + 1 divides 8^(2^n) + 1. All factors of generalized Fermat numbers F_n(a,b) := a^(2^n)+b^(2^n), a >= 2, n >= 0, are of the form k*2^m+1, k >= 1, m >=0 (Riesel (1994)). - Daniel Forgues, Jun 19 2011

			For n = 3, 8^(2^3) + 1 = 16777217. Similarly, (2^8)^3 + 1 = 16777217. Then 2^8 + 1 = 257 and 16777217/257 = 65281.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10
Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A000215 (Fermat numbers: 2^(2^n) + 1, n >= 0).

Cf. A059919, A199591, A078303, A078304, A080176, A199592, A152585.

[8^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Table[8^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

g(a,n) = if(a%2,b=2,b=1);for(x=0,n,y=a^(2^x)+b;print1(y","))

a(n)=1<<(3*2^n)+1 \\ Charles R Greathouse IV, Jul 29 2011

a(0)=9, a(n) = (a(n-1) - 1)^2 + 1, n >= 1.

Sum_{n>=0} 2^n/a(n) = 1/7. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 19 2011

A177888 P_n(k) with P_0(z) = z+1 and P_n(z) = z + P_(n-1)(z)*(P_(n-1)(z)-z) for n>1; square array P_n(k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 7, 1, 5, 7, 17, 43, 1, 6, 9, 31, 257, 1807, 1, 7, 11, 49, 871, 65537, 3263443, 1, 8, 13, 71, 2209, 756031, 4294967297, 10650056950807, 1, 9, 15, 97, 4691, 4870849, 571580604871, 18446744073709551617, 113423713055421844361000443, 1

Offset: 0

Alois P. Heinz, Dec 14 2010

			Square array P_n(k) begins:
  1,              2,          3,      4,       5,    6,    7,     8, ...
  1,              3,          5,      7,       9,   11,   13,    15, ...
  1,              7,         17,     31,      49,   71,   97,   127, ...
  1,             43,        257,    871,    2209, 4691, 8833, 15247, ...
  1,           1807,      65537, 756031, 4870849,  ...
  1,        3263443, 4294967297,    ...
  1, 10650056950807,        ...

Alois P. Heinz, Antidiagonals n = 0..13, flattened
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!)

Columns k=0-10 give: A000012, A000058(n+1), A000215, A000289(n+1), A000324(n+1), A001543(n+1), A001544(n+1), A067686, A110360(n+1), A110368(n+1), A110383(n+1).
Rows n=0-2 give: A000027(k+1), A005408, A056220(k+1).
Main diagonal gives A252730.
Coefficients of P_n(z) give: A177701.

p:= proc(n) option remember;
      z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
seq(seq(p(n)(d-n), n=0..d), d=0..8);

p[n_] := p[n] = Function[z, z + If [n == 0, 1, p[n-1][z]*(p[n-1][z]-z)] ]; Table [Table[p[n][d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A046052 Number of prime factors of Fermat number F(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5

Offset: 0

Eric W. Weisstein

F(12) has 6 known factors with C1133 remaining. [Updated by Walter Nissen, Apr 02 2010]
F(13) has 4 known factors with C2391 remaining.
F(14) has one known factor with C4880 remaining. [Updated by Matt C. Anderson, Feb 14 2010]
John Selfridge apparently conjectured that this sequence is not monotonic, so at some point a(n+1) < a(n). Related sequences such as A275377 and A275379 already exhibit such behavior. - Jeppe Stig Nielsen, Jun 08 2018
Factors are counted with multiplicity although it is unknown if all Fermat numbers are squarefree. - Jeppe Stig Nielsen, Jun 09 2018

W. Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
W. Keller, Summary of factoring status for Fermat numbersF(n)
PSI (The algorithm company), Fermat factor status [Broken link?]
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
Eric Weisstein's World of Mathematics, Fermat Number
Wikipedia, Selfridge's Conjecture about Fermat Numbers

Cf. A000215, A023394, A229850.

Array[PrimeOmega[2^(2^#) + 1] &, 9, 0] (* Michael De Vlieger, May 31 2022 *)

a(n)=bigomega(2^(2^n)+1) \\ Eric Chen, Jun 13 2018

a(n) = A001222(A000215(n)).

Name corrected by Arkadiusz Wesolowski, Oct 31 2011

cp's OEIS Frontend

A152585 Generalized Fermat numbers: 12^(2^n) + 1, n >= 0.

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A035089 Smallest prime of form 2^n*k + 1.

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A000289 A nonlinear recurrence: a(n) = a(n-1)^2 - 3*a(n-1) + 3 (for n>1).

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A050922 Triangle in which n-th row gives prime factors of n-th Fermat number 2^(2^n)+1.

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A121270 Prime Sierpinski numbers of the first kind: primes of the form k^k+1.

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A090872 a(n) is the smallest number m greater than 1 such that m^(2^k)+1 for k=0,1,...,n are primes.

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A000324 A nonlinear recurrence: a(0) = 1, a(1) = 5, a(n) = a(n-1)^2 - 4*a(n-1) + 4 for n>1.

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