A048736 -id:A048736 - cp's OEIS frontend

A208224 a(n)=(a(n-1)^2*a(n-3)^3+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

1, 1, 1, 1, 2, 5, 27, 5837, 2129410576, 17850077316687753782569, 2346851008195218976646246398770505953580095510848345967

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

This is the case a=3, b=1, c=2, y(0)=y(1)=y(2)=y(3)=1 of the recurrence shown in the Example 3.3 of "The Laurent phenomenon" (see Link lines, p. 10).

The next term (a(11)) has 133 digits. - Harvey P. Dale, Mar 06 2017

Seiichi Manyama, Table of n, a(n) for n = 0..13
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A048736, A208221, A208223, A208225, A208227.

y:=proc(n) if n<4 then return 1: fi: return (y(n-1)^2*y(n-3)^3+y(n-2))/y(n-4): end:
seq(y(n),n=0..11);

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]^2*a[n-3]^3+ a[n-2])/ a[n-4]},a,{n,10}] (* Harvey P. Dale, Mar 06 2017 *)

A208225 a(n)=(a(n-1)^3*a(n-3)^3+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 731, 3124943137, 11123050014071530610530827873034

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

This is the case a=3, b=1, c=3, y(0)=y(1)=y(2)=y(3)=1 of the recurrence shown in the Example 3.3 of "The Laurent phenomenon" (see Link lines, p. 10).

Seiichi Manyama, Table of n, a(n) for n = 0..10
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A048736, A208222, A208224, A208228.

y:=proc(n) if n<4 then return 1: fi: return (y(n-1)^3*y(n-3)^3+y(n-2))/y(n-4): end:
seq(y(n),n=0..9);

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]^3 a[n-3]^3+ a[n-2])/ a[n-4]},a,{n,10}] (* Harvey P. Dale, Aug 25 2016 *)

A208227 a(n) = (a(n-1)^2*a(n-3)^4+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 27, 11669, 42551737826, 192450770996317798484507077, 25433732883480327279167427243395261255488704554514737402263583619505

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

This is the case a=4, b=1, c=2, y(0)=y(1)=y(2)=y(3)=1 of the recurrence shown in the Example 3.3 of "The Laurent phenomenon" (see Link lines, p. 10).

Seiichi Manyama, Table of n, a(n) for n = 0..12
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A048736, A208224, A208226, A208228.

y:=proc(n) if n<4 then return 1: fi: return (y(n-1)^2*y(n-3)^4+y(n-2))/y(n-4): end:
seq(y(n),n=0..10);

a[n_]:=If[n<4,1, (a[n - 1]^2*a[n- 3]^4 + a[n - 2])/a[n - 4]]; Table[a[n], {n, 0, 10}] (* Indranil Ghosh, Mar 19 2017 *)

A208218 a(n)=(a(n-1)^2*a(n-3)+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 27, 1463, 5350936, 154615586811211, 1295349936263652139582251464117, 6137049788665571444030885529267632764941063995324839557922175605

Offset: 0

Matthew C. Russell, Apr 24 2012

nonn

This is the case a=1, b=1, c=2, y(0)=y(1)=y(2)=y(3)=1 of the recurrence shown in the Example 3.3 of "The Laurent phenomenon" (see Link lines, p. 10).

Seiichi Manyama, Table of n, a(n) for n = 0..14
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A048736.

[n le 4 select 1 else (Self(n-1)^2*Self(n-3)+Self(n-2))/Self(n-4): n in [1..12]]; // Bruno Berselli, Apr 24 2012

y:=proc(n) if n<4 then return 1: fi: return (y(n-1)^2*y(n-3)+y(n-2))/y(n-4): end:
seq(y(n),n=0..11);

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]^2*a[n-3]+ a[n-2])/ a[n-4]},a,{n,12}] (* Harvey P. Dale, Dec 25 2016 *)

A208221 a(n)=(a(n-1)^2*a(n-3)^2+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 27, 2921, 106653026, 1658455747832683945, 869174798276372512100586962107665002957113

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

This is the case a=2, b=1, c=2, y(0)=y(1)=y(2)=y(3)=1 of the recurrence shown in the Example 3.3 of "The Laurent phenomenon" (see Link lines, p. 10).

The next term (a(11)) has 97 digits. - Harvey P. Dale, Dec 17 2017

Seiichi Manyama, Table of n, a(n) for n = 0..13
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001); Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A048736, A208218, A208220, A208222, A208224.

y:=proc(n) if n<4 then return 1: fi: return (y(n-1)^2*y(n-3)^2+y(n-2))/y(n-4): end:
seq(y(n),n=0..11);

a[n_] := a[n] = If[n <= 3, 1, (a[n-1]^2*a[n-3]^2 + a[n-2])/a[n-4]];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 24 2017 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]^2 a[n-3]^2+ a[n-2])/ a[n-4]},a,{n,12}] (* Harvey P. Dale, Dec 17 2017 *)

A208223 a(n) = (a(n-1)*a(n-3)^3+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 43, 583, 24306, 386499545, 1781091354996947, 43869039083107828857967559, 104205727286975116465887590166696643681426291537, 1523355234093129576841463666274426784578547247551635338205747270819704358703763325458

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

This is the case a=3, b=1, c=1, y(0)=y(1)=y(2)=y(3)=1 of the recurrence shown in the Example 3.3 of "The Laurent phenomenon" (see Link lines, p. 10).

Seiichi Manyama, Table of n, a(n) for n = 0..18
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A048736.

[n le 4 select 1 else (Self(n-1)*Self(n-3)^3+Self(n-2))/Self(n-4): n in [1..15]]; // Bruno Berselli, Apr 26 2012

y:=proc(n) if n<4 then return 1: fi: return (y(n-1)*y(n-3)^3+y(n-2))/y(n-4): end:
seq(y(n),n=0..14);

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]a[n-3]^3+a[n-2])/ a[n-4]},a,{n,20}] (* Harvey P. Dale, Jul 13 2014 *)

A208228 a(n)=(a(n-1)^3*a(n-3)^4+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 731, 6249886265, 800859597553373777918076329400178

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

This is the case a=4, b=1, c=3, y(0)=y(1)=y(2)=y(3)=1 of the recurrence shown in the Example 3.3 of "The Laurent phenomenon" (see Link lines, p. 10).

Seiichi Manyama, Table of n, a(n) for n = 0..10
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A048736, A208225, A208227.

y:=proc(n) if n<4 then return 1: fi: return (y(n-1)^3*y(n-3)^4+y(n-2))/y(n-4): end:
seq(y(n),n=0..9);

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]^3 a[n-3]^4+ a[n-2])/ a[n-4]},a,{n,10}] (* Harvey P. Dale, Jan 08 2014 *)

A275175 a(n) = (2 * a(n-3) + a(n-1) * a(n-5)) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 5, 7, 13, 23, 83, 147, 215, 423, 771, 2801, 4971, 7281, 14351, 26181, 95133, 168845, 247317, 487493, 889373, 3231703, 5735737, 8401475, 16560393, 30212491, 109782751, 194846191, 285402811, 562565851, 1026335311, 3729381813, 6619034735, 9695294077, 19110678523

Offset: 0

Seiichi Manyama, Jul 19 2016

Inspired by A048736.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,35,0,0,0,0,-35,0,0,0,0,1).

Cf. A048736, A275173, A275176.

RecurrenceTable[{a[n] == (2 a[n - 3] + a[n - 1] a[n - 5])/a[n - 6], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 40}] (* Michael De Vlieger, Jul 19 2016 *)

Vec((1 +x +x^2 +x^3 +x^4 -34*x^5 -32*x^6 -30*x^7 -28*x^8 -22*x^9 +23*x^10 +13*x^11 +7*x^12 +5*x^13 +3*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -34*x^5 +x^10)) + O(x^50)) \\ Colin Barker, Jul 19 2016

def A(k, l, n)
  a = Array.new(k * 2, 1)
  ary = [1]
  while ary.size < n + 1
    break if (a[1] * a[-1] + a[k] * l) % a[0] > 0
    a = *a[1..-1], (a[1] * a[-1] + a[k] * l) / a[0]
    ary << a[0]
  end
  ary
end
def A275175(n)
  A(3, 2, n)
end

G.f.: (1 +x +x^2 +x^3 +x^4 -34*x^5 -32*x^6 -30*x^7 -28*x^8 -22*x^9 +23*x^10 +13*x^11 +7*x^12 +5*x^13 +3*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -34*x^5 +x^10)). - Colin Barker, Jul 19 2016

a(n) = 35*a(n-5) - 35*a(n-10) + a(n-15). - G. C. Greubel, Jul 20 2016

A275176 a(n) = (3 * a(n-3) + a(n-1) * a(n-5)) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 7, 10, 22, 43, 202, 370, 547, 1264, 2521, 11881, 21781, 32221, 74521, 148681, 700744, 1284667, 1900450, 4395442, 8769643, 41331982, 75773530, 112094287, 259256524, 517260241, 2437886161, 4469353561, 6611662441, 15291739441, 30509584561

Offset: 0

Seiichi Manyama, Jul 19 2016

Inspired by A048736.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,60,0,0,0,0,-60,0,0,0,0,1).

Cf. A048736, A275173, A275175.

RecurrenceTable[{a[n] == (3 a[n - 3] + a[n - 1] a[n - 5])/a[n - 6], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 36}] (* Michael De Vlieger, Jul 19 2016 *)

Vec((1 +x +x^2 +x^3 +x^4 -59*x^5 -56*x^6 -53*x^7 -50*x^8 -38*x^9 +43*x^10 +22*x^11 +10*x^12 +7*x^13 +4*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -59*x^5 +x^10)) + O(x^50)) \\ Colin Barker, Jul 19 2016

def A(k, l, n)
  a = Array.new(k * 2, 1)
  ary = [1]
  while ary.size < n + 1
    break if (a[1] * a[-1] + a[k] * l) % a[0] > 0
    a = *a[1..-1], (a[1] * a[-1] + a[k] * l) / a[0]
    ary << a[0]
  end
  ary
end
def A275176(n)
  A(3, 3, n)
end

G.f.: (1 +x +x^2 +x^3 +x^4 -59*x^5 -56*x^6 -53*x^7 -50*x^8 -38*x^9 +43*x^10 +22*x^11 +10*x^12 +7*x^13 +4*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -59*x^5 +x^10)). - Colin Barker, Jul 19 2016

a(n) = 60*a(n-5) - 60*a(n-10) + a(n-15).

A276531 a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-3) * a(n-4)) / a(n-6), with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 5, 11, 41, 247, 1498, 39629, 3121233, 1344630757, 4527359876765, 673384475958949877, 12684198948982702826816701, 103442271685605704255863097581658042, 12389248756108266360505757651017660004796444483503, 657084395567781339286109602463271066924826185667810218784212689809097

Offset: 0

Seiichi Manyama, Nov 16 2016

nonn

This sequence is the generalization of Dana Scott's sequence (A048736).
Conjecture: a(n) is an integer for all n. It has been checked by computer for 0 <= n <= 50.
The recursion has the Laurent property. If a(0), ..., a(5) are variables, then a(n) is a Laurent polynomial (a rational function with a monomial denominator). - Michael Somos, Nov 21 2016

Seiichi Manyama, Table of n, a(n) for n = 0..28

Cf. A048736, A006721.

a:=[1,1,1,1,1,1];; for n in [7..25] do a[n]:=(a[n-1]*a[n-5]+a[n-2]*a[n-3]*a[n-4])/a[n-6]; od; a; # Muniru A Asiru, Jul 30 2018

I:=[1,1,1,1,1,1]; [n le 6 select I[n] else (Self(n-1)*Self(n-5) + Self(n-2)*Self(n-3)*Self(n-4))/Self(n-6): n in [1..30]]; // G. C. Greubel, Jul 30 2018

RecurrenceTable[{a[n] == (a[n - 1] a[n - 5] + a[n - 2] a[n - 3] a[n - 4])/a[n - 6], a[0] == a[1] == a[2] == a[3] == a[4] == a[5] == 1}, a, {n, 0, 21}] (* Michael De Vlieger, Nov 21 2016 *)
nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,(b*f+d*e*c)/a}; NestList[nxt,{1,1,1,1,1,1},30][[All,1]] (* Harvey P. Dale, Nov 21 2021 *)

def A(k, n)
  a = Array.new(k, 1)
  ary = [1]
  while ary.size < n + 1
    i = a[-1] * a[1] + a[2..-2].inject(:*)
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A276531(n)
  A(6, n)
end

a(n) * a(n-6) = a(n-1) * a(n-5) + a(n-2) * a(n-3) * a(n-4).

a(5-n) = a(n) for all n in Z.

cp's OEIS Frontend

A208224 a(n)=(a(n-1)^2*a(n-3)^3+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

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A208225 a(n)=(a(n-1)^3*a(n-3)^3+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

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A208227 a(n) = (a(n-1)^2*a(n-3)^4+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

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A208218 a(n)=(a(n-1)^2*a(n-3)+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

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A208221 a(n)=(a(n-1)^2*a(n-3)^2+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

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A208223 a(n) = (a(n-1)*a(n-3)^3+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

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A208228 a(n)=(a(n-1)^3*a(n-3)^4+a(n-2))/a(n-4) with a(0)=a(1)=a(2)=a(3)=1.

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A275175 a(n) = (2 * a(n-3) + a(n-1) * a(n-5)) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

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