A258113 -id:A258113 - 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

A076949 Decimal expansion of c, the constant such that lim n -> infinity A003095(n)/c^(2^n) = 1.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Nov 27 2002

			1.2259024435287485386279474959130085213212293209696612823177009072552339975...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Stephan Wagner, Volker Ziegler, Irrationality of growth constants associated with polynomial recursions, arXiv:2004.09353 [math.NT], 2020.

Cf. A003095, A077496, A213437, A258112, A258113.

function A003095(n)
  if n eq 0 then return 0;
  else return 1 + A003095(n-1)^2;
  end if; return A003095;
end function;
function S(n)
  if n eq 1 then return Log(2)/2;
  else return S(n-1) + Log(1 + 1/A003095(n)^2)/2^n;
  end if; return S;
end function;
SetDefaultRealField(RealField(120)); Exp(S(12)/2); // G. C. Greubel, Nov 29 2022

A003095[n_]:= A003095[n]= If[n==0, 0, 1 + A003095[n-1]^2];
S[n_]:= S[n]= If[n==1, Log[2]/2, S[n-1] + Log[1 + 1/A003095[n]^2]/2^n];
RealDigits[Exp[S[13]/2], 10, 120][[1]] (* G. C. Greubel, Nov 29 2022 *)

@CachedFunction
def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
@CachedFunction
def S(n): return log(2)/2 if (n==1) else S(n-1) + log(1 + 1/(A003095(n))^2)/2^n
numerical_approx( exp(S(12)/2), digits=120) # G. C. Greubel, Nov 29 2022

Equals sqrt(A077496). - Vaclav Kotesovec, Dec 17 2014

A250309 a(n) = a(n-1)*(1 + a(n-1)/a(n-3)), with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 6, 42, 924, 143220, 488523420, 258285263294520, 465795819523189050504840, 444125576385425970712647062585372630520, 763680920404535561780141108036287312478667174369871222219397040

Offset: 0

Michael Somos, Jan 16 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..17

Cf. A007660, A258113.

I:=[1,1,1]; [n le 3 select I[n] else Self(n-1)*(1 + Self(n-1)/Self(n-3)): n in [1..15]]; // G. C. Greubel, Aug 03 2018

RecurrenceTable[{a[n]==a[n-1]*(1 + a[n-1]/a[n-3]), a[0]==1, a[1]==1, a[2]==1},a,{n,0,12}] (* Vaclav Kotesovec, Jan 18 2015 *)

{a(n) = if( n<3, n>=0, a(n-1)*(1 + a(n-1)/a(n-3)))};

0 = a(n)*(a(n+2) - a(n+3)) + a(n+2)*a(n+2) for all n>=0.
A007660(n+1) = a(n)/a(n-1).
a(n) ~ b * f^(d^n), where b = 0.270887790039424376..., f = c^(2+sqrt(5)) = 1.574173161904651669837597516422779594... and d = (1+sqrt(5))/2. For the constant c = A258113 = 1.11305797590293193285359770716758491... see A007660. - Vaclav Kotesovec, Jan 18 2015

A258112 Decimal expansion of a constant related to A001056.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 20 2015

			1.7978784900091604813559508837031352161793665265019525368552362542745...

Cf. A001056, A076949, A258113.

A001056 = RecurrenceTable[{a[0]==1, a[1]==N[3, 200], a[n] == a[n-1]*a[n-2]+1}, a[n], {n, 1, 30}]; Do[Print[N[Exp[c2]/.Solve[Table[Log[A001056[[n]]] == c1*((1-Sqrt[5])/2)^n + c2*((1+Sqrt[5])/2)^n, {n, k, k+1}]], 120][[1]]], {k, Length[A001056]-2, Length[A001056]-1}];

Equals limit n->infinity (A001056(n))^((2/(1+sqrt(5)))^n).

cp's OEIS Frontend

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

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A076949 Decimal expansion of c, the constant such that lim n -> infinity A003095(n)/c^(2^n) = 1.

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A250309 a(n) = a(n-1)*(1 + a(n-1)/a(n-3)), with a(0) = a(1) = a(2) = 1.

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A258112 Decimal expansion of a constant related to A001056.

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