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A112373 a(n+2) = (a(n+1)^3 + a(n+1)^2)/a(n) with a(0)=1, a(1)=1. 10
1, 1, 2, 12, 936, 68408496, 342022190843338960032, 584861200495456320274313200204390612579749188443599552 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

As n tends to infinity, log(log(a(n)))/n tends to log((3+sqrt(5))/2) = A104457.

The Laurent property is satisfied by any second-order recurrence of the form a(n+2)=f(a(n+1))/a(n) where f is a polynomial of the form f(x)=x^m*p(x) with m a positive integer greater than or equal to 2 and p arbitrary. In that case a(0)=a(1)=1 generates a sequence of integers and the ratios a(n+1)/a(n) and a(n+1)*a(n-1)/a(n)^2 are integers for all n. - Andrew Hone, Dec 12 2005

Also denominators of sum(1/a(k)): k=0..n) with numerators = A259644. - Reinhard Zumkeller, Jul 02 2015

The next term (a(8)) has 141 digits. - Harvey P. Dale, Apr 05 2019

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10

Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.

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

Andrew N. W. Hone, Curious continued fractions, nonlinear recurrences and transcendental numbers, arXiv:1507.00063 [math.NT], 2015, and Journal of Integer Sequences 18 (2015), Article #15.8.4.

A. N. W. Hone, Continued fractions for some transcendental numbers, arXiv:1509.05019 [math.NT], 2015-2016, Monatsh. Math. [DOI].

FORMULA

a(n+1) / a(n) = A114552(n), a(n) = a(1-n) for all n in Z. - Michael Somos, Apr 19 2017

MAPLE

a[0]:=1; a[1]:=1; f(x):=x^3+x^2; for n from 0 to 8 do a[n+2]:=simplify(subs(x=a[n+1], f(x))/a[n]) od; s[3]:=ln(2^2*3); s[4]:=ln(2^3*3^2*13); for n from 3 to 10000 do s[n+2]:=evalf(3*s[n+1]+ln(1+exp(-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...

MATHEMATICA

RecurrenceTable[{a[n] == (a[n - 1]^3 + a[n - 1]^2)/a[n - 2], a[0] == a[1] == 1}, a, {n, 0, 7}] (* Michael De Vlieger, Jul 02 2015 *)

nxt[{a_, b_}]:={b, (b^3+b^2)/a}; NestList[nxt, {1, 1}, 8][[All, 1]] (* Harvey P. Dale, Apr 05 2019 *)

PROG

(MAGMA) I:=[1, 1]; [n le 2 select I[n] else (Self(n-1)^3+Self(n-1)^2)/Self(n-2): n in [1..10]]; // Vincenzo Librandi, Jul 02 2015

(Haskell)

a112373 n = a112373_list !! n

a112373_list = 1 : 1 : zipWith (\u v -> (u^3 + u^2) `div` v)

                               (tail a112373_list) a112373_list

-- Reinhard Zumkeller, Jul 02 2015

(PARI) {a(n) = my(a=self()); if(n<0, a(1-n), n<2, 1, a(n-1)^2 * (1 + a(n-1)) / a(n-2))}; /* Michael Somos, Apr 19 2017 */

CROSSREFS

Cf. A114551, A114552.

Cf. A259644.

Sequence in context: A222207 A129933 A064320 * A058975 A057120 A112512

Adjacent sequences:  A112370 A112371 A112372 * A112374 A112375 A112376

KEYWORD

nonn

AUTHOR

Andrew Hone, Dec 02 2005

STATUS

approved

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Last modified July 12 16:03 EDT 2020. Contains 335665 sequences. (Running on oeis4.)