A061021 -id:A061021 - cp's OEIS frontend

A022405 a(n) = a(n-1)*a(n-2) - a(n-3), with a(1) = 0, a(2) = 1, a(3) = 2.

0, 1, 2, 2, 3, 4, 10, 37, 366, 13532, 4952675, 67019597734, 331926286207224918, 22245566178948766568816183137, 7383888166355511098764350563784314022618210032

Offset: 1

Robert G. Wilson v, Jul 05 2000

Theorem 1.1 of Hare et al. (2010, 2011) involves a shifted version of this sequence and the Fibonacci sequence A000045. (The program by Alonso del Arte below does involve a shifted version of this sequence.) - Petros Hadjicostas, May 11 2019

G. C. Greubel, Table of n, a(n) for n = 1..21
Kevin G. Hare, Ian D. Morris, Nikita Sidorov, and Jacques Theys, An explicit counterexample to the Lagarias-Wang finiteness conjecture, arXiv:1006.2117 [math.OC], 2010-2011.
Kevin G. Hare, Ian D. Morris, Nikita Sidorov, and Jacques Theys, An explicit counterexample to the Lagarias-Wang finiteness conjecture, Advances in Mathematics 226 (2011), 4667-4701.

Cf. A001622, A061021, A061292, A072878, A072879, A072880, A178768.

I:=[0,1,2]; [n le 3 select I[n] else Self(n-1)*Self(n-2) - Self(n-3): n in [1..15]];  // G. C. Greubel, Mar 01 2018

a[1] = 0; a[2] = 1; a[3] = 2; a[n_] := a[n] = a[n - 1] a[n - 2] - a[n - 3]; Table[a[n], {n, 1, 15}] (* Alonso del Arte, Jan 31 2011 *)
nxt[{a_,b_,c_}]:={b,c,c*b-a}; NestList[nxt,{0,1,2},15][[;;,1]] (* Harvey P. Dale, Mar 23 2025 *)

It appears that lim_{n->infinity} log(a(n))/phi^n = 0.07743008049000107520747623421744398272089261907514..., where phi = (1 + sqrt(5))/2 is the golden ratio A001622. - Petros Hadjicostas and Jon E. Schoenfield, May 11 2019

Name clarified by Michel Marcus, May 10 2019

A061292 a(n) = a(n-1)a(n-2)a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.

Original entry on oeis.org

2, 2, 2, 2, 6, 22, 262, 34582, 199330642, 1806032092550706, 12449434806576800059248920402, 4481765860945171681908664776799089162954814190172722

Offset: 0

Stephen G Penrice, Jun 04 2001

Any four consecutive terms are a solution to the Diophantine equation w^2 + x^2 + y^2 + z^2 = wxyz.

a(n) = 2 * A072878(n+1).

Vincenzo Librandi, Table of n, a(n) for n = 0..18
Michael Magee and Brady Haran, A simple equation that behaves weirdly, Numberphile, Youtube video, 2025.

Cf. A022405, A061021, A072878, A072879, A072880.

a061292 n = a061292_list !! n
a061292_list = 2 : 2 : 2 : 2 : zipWith (-)
   (zipWith3 (((*) .) . (*)) (drop 2 xs) (tail xs) xs) a061292_list
   where xs = tail a061292_list
-- Reinhard Zumkeller, Mar 25 2015

I:=[2,2,2,2]; [n le 4 select I[n] else Self(n-1)*Self(n-2)*Self(n-3)-Self(n-4): n in [1..12]]; // Vincenzo Librandi, Sep 17 2011

a[1] := 2; a[2] := 2; a[3] := 2; a[4] := 2; a[n_] := a[n - 1]*a[n - 2]*a[n - 3] - a[n - 4]; Table[a[n], {n, 1, 15}] (* Stefan Steinerberger, Mar 31 2006 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==2,a[n]==a[n-1]a[n-2]a[n-3]- a[n-4]},a[n],{n,12}] (* Harvey P. Dale, Sep 15 2011 *)

More terms from Larry Reeves (larryr(AT)acm.org) and Jason Earls, Jun 05 2001

A074394 a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.

Original entry on oeis.org

1, 2, 3, 5, 13, 62, 801, 49649, 39768787, 1974480504962, 78522694637486171445, 155041529758800625329015665441303, 12174278697379026530632791354719900462885271361687873

Offset: 1

Henry Bottomley, Sep 24 2002

nonn

All consecutive quadruples are pairwise coprime. Multiples of 2 occur when n=2 mod 4, multiples of 3 when n=3 mod 4, multiples of 5 when n=4 mod 7, multiples of 7 when n=10 mod 14, multiples of 9 when n=7 or 11 mod 24, multiples of 10 when n=18 mod 28. Multiples of 4, 6 and 8 never occur.

			a(6) = a(5)*a(4) - a(3) = 13*5 - 3 = 62.

Reinhard Zumkeller, Table of n, a(n) for n = 1..19

Cf. A001622, A022405, A061021, A072878, A072879, A072880.

a074394 n = a074394_list !! (n-1)
a074394_list = 1 : 2 : 3 : zipWith (-)
   (tail $ zipWith (*) (tail a074394_list) a074394_list) a074394_list
-- Reinhard Zumkeller, Mar 25 2015

nxt[{a_,b_,c_}]:={b,c,b*c-a}; NestList[nxt,{1,2,3},15][[All,1]] (* Harvey P. Dale, Jan 21 2023 *)

Lim_{n->infinity} a(n+1)/a(n)^phi = 1, where phi is the golden ratio (1+sqrt(5))/2 = A001622. - Benoit Cloitre, Sep 26 2002
From Jon E. Schoenfield, May 13 2019: (Start)
It appears that, for n >= 2,
  a(n) = ceiling(e^(c*phi^n - d/(-phi)^n))
where
  phi = (1 + sqrt(5))/2
    c = 0.230193077518834725477008740044380256486365499661...
    d = 0.067704372842879037264190305626317036100889750046...
(End)

cp's OEIS Frontend

A022405 a(n) = a(n-1)*a(n-2) - a(n-3), with a(1) = 0, a(2) = 1, a(3) = 2.

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A061292 a(n) = a(n-1)a(n-2)a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.

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A074394 a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.

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cp's OEIS Frontend

A022405 a(n) = a(n-1)*a(n-2) - a(n-3), with a(1) = 0, a(2) = 1, a(3) = 2.

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Magma

Mathematica

Formula

Extensions

A061292 a(n) = a(n-1)*a(n-2)*a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.

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Haskell

Magma

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Extensions

A074394 a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.

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Haskell

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Formula

A061292 a(n) = a(n-1)a(n-2)a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.