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A112449 a(n+2) = (a(n+1)^3 + a(n+1))/a(n) with a(0)=1, a(1)=1. 1
1, 1, 2, 10, 505, 12878813, 4229958765311886322, 5876687051603582015287706866081267480733704277890 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A second-order recurrence with the Laurent property. This property is satisfied by any second-order recurrence of the form a(n+2) = f(a(n+1))/a(n) with f being a polynomial of the form f(x) = x*p(x) where p is a polynomial of degree d with integer coefficients such that p(0)=1 and p has the reciprocal property x^d*p(1/x) = p(x). Hence if a(0) = a(1) = 1 then a(n) is an integer for all n.

As n tends to infinity, log(log(a(n)))/n tends to log((3+sqrt(5))/2) or about 0.962 (A202543).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10

S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.

FORMULA

a(1-n) = a(n). - Seiichi Manyama, Nov 20 2016

MAPLE

a[0]:=1; a[1]:=1; f(x):=x^3+x;

for n from 0 to 8 do a[n+2]:=simplify(subs(x=a[n+1], f(x))/a[n]) od;

s[3]:=ln(10); s[4]:=ln(505);

for n from 3 to 10000 do s[n+2]:=evalf(3*s[n+1]+ln(1+exp(-2*s[n+1]))-s[n]): od: print(evalf(ln(s[10002])/(10002))): evalf(ln((3+sqrt(5))/2));

# s[n]=ln(a[n]); ln(s[n])/n converges slowly to 0.962...

f:=proc(n) option remember; local i, j, k, t1, t2, t3; if n <= 1 then RETURN(1); fi; (f(n-1)^3+f(n-1))/f(n-2); end;

# N. J. A. Sloane

MATHEMATICA

nxt[{a_, b_}]:={b, (b^3+b)/a}; NestList[nxt, {1, 1}, 10][[All, 1]] (* Harvey P. Dale, Jun 26 2017 *)

PROG

(Ruby)

def A(l, m, n)

  a = Array.new(2 * m, 1)

  ary = [1]

  while ary.size < n + 1

    i = a[1..-1].inject(:*) + a[m] ** l

    break if i % a[0] > 0

    a = *a[1..-1], i / a[0]

    ary << a[0]

  end

  ary

end

def A112449(n)

  A(3, 1, n)

end # Seiichi Manyama, Nov 20 2016

CROSSREFS

Cf. A101879, A112373, A202543.

Sequence in context: A059723 A334286 A265627 * A011824 A064300 A290060

Adjacent sequences:  A112446 A112447 A112448 * A112450 A112451 A112452

KEYWORD

nonn

AUTHOR

Andrew Hone, Dec 12 2005

STATUS

approved

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Last modified June 1 20:49 EDT 2020. Contains 334765 sequences. (Running on oeis4.)