A022405 -id:A022405 - cp's OEIS frontend

A166038 Period of A022405 mod n.

1, 3, 12, 18, 30, 12, 42, 18, 24, 30, 48, 36, 42, 42, 60, 36, 102, 24, 78, 90, 84, 48, 24, 36, 30, 42, 72, 126, 84, 60, 312, 72, 48, 102, 210, 72, 84, 78, 84, 90, 48, 84, 66, 144, 120, 24, 18, 36, 42, 30, 204, 126, 276, 72, 240, 126, 156, 84, 144, 180, 444, 312, 168, 144

Offset: 1

Carl R. White, Oct 05 2009

Pisano-style numbers for the recurrence b(n) = n for 0 <= n <= 2, otherwise b(n) = b(n-1)*b(n-2) - b(n-3); that is, the sequence b(n) = A022405(n+1).

Eric Weisstein's World of Mathematics, Pisano period.
Wikipedia, Pisano period.

Cf. A001175, A022405, A046738.

/* GNU bc */ define p(m){auto os,a,b,c,d,k;if(m<3)return 2*m-1;os=scale;scale=0;a=0;b=1;c=2;k=0;while(1){d=(c*b+m-a)%m;a=b;b=c;c=d;k+=1;if(a==0&&b==1&&c==2)break};scale=os;return k};for(i=1;i<=200;i++)print p(i),",";print"\n"

A072878 a(n) = 4a(n-1)a(n-2)*a(n-3) - a(n-4) with a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 11, 131, 17291, 99665321, 903016046275353, 6224717403288400029624460201, 2240882930472585840954332388399544581477407095086361, 50384188378657848181032338163962292285660644698840136656562636145266593550842871302412156442811

Offset: 1

Benoit Cloitre, Jul 28 2002

A subsequence of the generalized Markoff numbers.

Seiichi Manyama, Table of n, a(n) for n = 1..16
Arthur Baragar, Integral solutions of the Markoff-Hurwitz equations, J. Number Theory 49 (1994), 27-44.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, arXiv:math/0601324 [math.NT], 2006.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, J. Phys. A: Math. Gen. 39 (2006), L171-L177.
Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrence, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

Cf. A022405, A064098, A075276, A072879, A072880.

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==4a[n-1]a[n-2]a[n-3]-a[n-4]},a,{n,15}] (* Harvey P. Dale, Nov 29 2014 *)

a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)^2 + a(n-3)^2 + a(n-2)^2)/a(n-4) for n >= 5.

From the recurrence a(n) = 4*a(n-1)*a(n-2)*a(n-3) - a(n-4), any four successive terms satisfy the Markoff-Hurwitz equation a(n)^2 + a(n-1)^2 + a(n-2)^2 + a(n-3)^2 = 4*a(n)*a(n-1)*a(n-2)*a(n-3), cf. A075276. As n tends to infinity, the limit of log(log(a(n)))/n is log x = 0.6093778633..., where x=1.839286755... is the real root of the cubic x^3 - x^2 - x - 1 = 0. - Andrew Hone, Nov 14 2005

Entry revised Nov 19 2005, based on comments from Andrew Hone
a(13) from Harvey P. Dale, Nov 29 2014
Name clarified by Petros Hadjicostas, May 11 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

A072877 a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)*a(n-3) + a(n-2)^4)/a(n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 19, 119, 65339, 67258454, 959259994615659593, 171965197021698738644442682357, 12959040525296547835480490169418622922155526267774117749963303914461

Offset: 1

Benoit Cloitre, Jul 28 2002

A variation of a Somos-4 sequence with a(n-2)^4 in place of a(n-2)^2.

Seiichi Manyama, Table of n, a(n) for n = 1..17
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), pp. 589-633.
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics 28 (2002), pp. 119-144.
David Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, arXiv:math/0601324 [math.NT], 2006.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, J. Phys. A: Math. Gen. 39 (2006), pp. L171-L177.

Cf. A006720, A022405, A061292, A065918, A072878, A072879, A072880, A074394, A178768.

L[0]:=0; L[1]:=0; L[2]:=0; L[3]:=0; for n from 0 to 4000 do L[n+4]:=evalf(ln(1+exp(L[n+3]+L[n+1]-4*L[n+2]))+4*L[n+2]-L[n]): od: for n from 3990 to 4000 do print(evalf(ln(L[n+4])/(n+4))): od: #Note: L[n] is log(a[n]) # Andrew Hone, Nov 15 2005

nxt[{a_,b_,c_,d_}]:={b,c,d,(d*b+c^4)/a}; NestList[nxt,{1,1,1,1},15][[All,1]] (* Harvey P. Dale, Jun 01 2022 *)

Lim_{n->infinity} (log(log(a(n))))/n = log(2+sqrt(3))/2 = A065918/2 or about 0.658. - Andrew Hone, Nov 15 2005; corrected by Michel Marcus, May 12 2019
From Jon E. Schoenfield, May 12 2019: (Start)
It appears that, for n >= 1,
  a(n) = ceiling(e^(c0*x^n + d0/x^n)) if n is even,
         ceiling(e^(c1*x^n + d1/x^n)) if n is odd,
where
  x  = sqrt(2 + sqrt(3)) = (sqrt(2) + sqrt(6))/2
  c0 =   0.024915247166055931001426396817534982995670642690...
  c1 =   0.029604794868229453467890216788323427656809346011...
  d0 = -10.535089427608481105514469573411011428431309483956...
  d1 =  -2.856773870202800001336732759121362374871088274450...
(End)

Definition corrected by Matthew C. Russell, Apr 24 2012

A061021 a(n) = a(n-1)*a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 3.

Original entry on oeis.org

3, 3, 3, 6, 15, 87, 1299, 112998, 146784315, 16586334025071, 2434613678231239448367, 40381315689150066251526220641224742, 98312903521778500654864668915856114278134197773017871243

Offset: 0

Stephen G Penrice, May 23 2001

Any three consecutive terms are a solution to the Diophantine equation x^2 + y^2 + z^2 = xyz.

Harry J. Smith, Table of n, a(n) for n = 0..17
Loren C. Larson, Solution to Problem Proposal 701, College Mathematics Journal 33 (2002), pp. 241-242.
Edward T. H. Wang, Problem Proposal 701, College Mathematics Journal 32 (2001), p. 211.

Cf. A022405, A061292, A072878, A072879, A072880, A074394, A178768.

a061021 n = a061021_list !! n
a061021_list = 3 : 3 : 3 : zipWith (-)
(tail $ zipWith (*) (tail a061021_list) a061021_list) a061021_list
-- Reinhard Zumkeller, Mar 25 2015

RecurrenceTable[{a[n] == a[n - 1] a[n - 2] - a[n - 3], a[0] == a[1] == a[2] == 3}, a, {n, 0, 12}] (* Michael De Vlieger, Aug 21 2016 *)

for (n=0, 17, if (n>2, a=a1*a2 - a3; a3=a2; a2=a1; a1=a, if (n==0, a=a3=3, if (n==1, a=a2=3, a=a1=3))); write("b061021.txt", n, " ", a)) \\ Harry J. Smith, Jul 16 2009

From Jon E. Schoenfield, May 12 2019: (Start)
It appears that, for n >= 1,
  a(n) = ceiling(e^(c0*phi^n - c1/(-phi)^n))
where
  phi = (1 + sqrt(5))/2,
   c0 = 0.4004033011137849744572073756789830081726425559860...
   c1 = 0.2798639753144007577581523025628820390768226527315...
(End)

More terms from Erich Friedman, Jun 03 2001

Name clarified by Petros Hadjicostas, May 11 2019

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)

A178768 Decimal expansion of real constant in an explicit counterexample to the Lagarias-Wang finiteness conjecture.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Jun 11 2010

			0.74932654633036755794396194809134467209132737023606431735802...

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.
J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices, Linear Algebra Appl., 214 (1995), 17-42.

Cf. A000045, A022405.

t(n) = if (n==0, 1, if (n==1, 2, if (n==2, 2, t(n-1)*t(n-2) - t(n-3)))); \\ A022405
prodinf(n=1, (1 - t(n-1)/(t(n)*t(n+1)))^((-1)^n*fibonacci(n+1))) \\ Michel Marcus, Jun 14 2015; May 10 2019

Equals Product_{n >= 1} (1 - t(n-1)/(t(n)*t(n+1)))^((-1)^n*Fibonacci(n+1)), where t(n) = A022405(n+1) and Fibonacci(n) = A000045(n). See Theorem 1.1 of Hare et al. (2010, 2011). - Michel Marcus, May 10 2019

More terms from Amiram Eldar, May 15 2021

cp's OEIS Frontend

A166038 Period of A022405 mod n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

bc

A072878 a(n) = 4*a(n-1)*a(n-2)*a(n-3) - a(n-4) with a(1) = a(2) = a(3) = a(4) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Extensions

A072877 a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)*a(n-3) + a(n-2)^4)/a(n-4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A061021 a(n) = a(n-1)*a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A178768 Decimal expansion of real constant in an explicit counterexample to the Lagarias-Wang finiteness conjecture.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A072878 a(n) = 4a(n-1)a(n-2)*a(n-3) - a(n-4) with a(1) = a(2) = a(3) = a(4) = 1.

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.