A250309 -id:A250309 - cp's OEIS frontend

A007660 a(n) = a(n-1)*a(n-2) + 1 with a(0) = a(1) = 0.

0, 0, 1, 1, 2, 3, 7, 22, 155, 3411, 528706, 1803416167, 953476947989903, 1719515742866809222961802, 1639518622529236077952144318816050685207, 2819178082162327154499022366029959843954512194276761760087463015

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

If we omit the first three terms of the sequence, a(n)/a(n-1) can be expressed as the continued fraction [a(n-2); a(n-1)]. - Eric Angelini, Feb 10 2005
This may be regarded as a multiplicative dual of the Fibonacci sequence A000045. Write Fibonacci's formula as F(0)=0, F(1)=1; F(n)=[F(n-1)+F(n-2)]*1 with n>1. Swap '+' and '*' and we have the present sequence! - B. Joshipura (bhushit(AT)yahoo.com), Aug 29 2007
a(n+1) divides a(2n+1), a(3n+1), a(4n+1), etc., this is because modulo a(n+1): a(1)=a(n+1)=0 and a(2)=a(n+2)=1 so the sequence repeats modulo a(n+1) with period n. - Isaac Kaufmann, Sep 04 2020

			b(10) / b(5) = 1803416167 / 7 = 257630881. - _Michael Somos_, Dec 29 2012

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

Vincenzo Librandi, Table of n, a(n) for n = 0..19 (shortened by _N. J. A. Sloane_, Jan 13 2019)
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!)
Bhushit Joshipura, 1, 1, 2, 3, 7, ... Multiplication dual of Fibonacci?, posting in newsgroup sci.math, Jul 28 2007.
S. Kak, The Golden Mean and the Physics of Aesthetics, arXiv:physics/0411195 [physics.hist-ph], 2004.

Cf. A250309, A253853, A258113.

a007660 n = a007660_list !! n
a007660_list = 0 : 0 : map (+ 1)
                       (zipWith (*) a007660_list $ tail a007660_list)
-- Reinhard Zumkeller, Jan 17 2015

I:=[0,0]; [n le 2 select I[n] else Self(n-1)*Self(n-2)+1: n in [1..20]]; // Vincenzo Librandi, Nov 14 2011

a[0] = a[1] = 0; a[n_] := a[n - 1]*a[n - 2] + 1; Table[ a[n], {n, 0, 15} ]
RecurrenceTable[{a[0]==a[1]==0,a[n]==a[n-1]a[n-2]+1},a,{n,20}] (* Harvey P. Dale, Nov 12 2011 *)

a(n) := if (n=0 or n=1) then 0 else a(n-1)*a(n-2)+1 $
makelist(a(n),n,0,18); /* Emanuele Munarini, Mar 24 2017 */

a(n) is asymptotic to c^(phi^n) where phi = (1 + sqrt(5))/2 and c = A258113 = 1.1130579759029319... - Benoit Cloitre, Sep 26 2003

b(n) = a(n+1) is a divisibility sequence. - Michael Somos, Dec 29 2012

A258113 Decimal expansion of a constant related to A007660.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 20 2015

			1.1130579759029319328535977071675849190660018151018652720143797242069...

Cf. A007660, A076949, A250309, A258112.

A007660 = RecurrenceTable[{a[1]==0, a[2]==N[1,200], a[n]==a[n-1]*a[n-2]+1},a[n],{n,1,30}]; Do[Print[N[Exp[c2]/.Solve[Table[Log[A007660[[n]]]==c1*((1-Sqrt[5])/2)^n + c2*((1+Sqrt[5])/2)^n, {n,k,k+1}]], 120][[1]]],{k, Length[A007660]-2, Length[A007660]-1}];

Equals limit n->infinity (A007660(n))^((2/(1+sqrt(5)))^n).

A276416 a(n) = a(n-1)*(1 + a(n-1)/a(n-4)), with a(0) = a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 42, 1806, 1632624, 444245153520, 4698898962968253924720, 12225720633546031105793020748137513851120, 91550929674875028299231929179221527919681972461210779957660001348767546720

Offset: 0

Seiichi Manyama, Sep 02 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..16

Cf. A133400, A250309.

RecurrenceTable[{a[n] == a[n - 1] (1 + a[n - 1]/a[n - 4]), a[0] == a[1] == a[2] == a[3] == 1}, a, {n, 0, 12}] (* Michael De Vlieger, Sep 04 2016 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,d(1+d/a)}; NestList[nxt,{1,1,1,1},12][[;;,1]] (* Harvey P. Dale, May 08 2025 *)

def A(m, n)
  a = Array.new(m, 1)
  ary = [1]
  while ary.size < n + 1
    break if a[-1] % a[0] > 0
    a = *a[1..-1], a[-1] * (1 + a[-1] / a[0])
    ary << a[0]
  end
  ary
end
def A276416(n)
  A(4, n)
end

0 = a(n)*(a(n+3) - a(n+4)) + a(n+3)*a(n+3) for all n>=0.

A133400(n) = a(n+1)/a(n).

cp's OEIS Frontend

A007660 a(n) = a(n-1)*a(n-2) + 1 with a(0) = a(1) = 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Maxima

Formula

A258113 Decimal expansion of a constant related to A007660.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A276416 a(n) = a(n-1)*(1 + a(n-1)/a(n-4)), with a(0) = a(1) = a(2) = a(3) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Ruby

Formula