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A100441 a(n) is the denominator of f(n) where f(1) = 2 and f(n+1) is the solution of x + Sum_{i=1..n} f(i) = x * Product_{i=1..n} f(i). 10
1, 1, 3, 13, 217, 57073, 3811958497, 16605534578235736513, 309708098978072051970763989442580255617, 106322990835084829467725909226560893968664147958670035553130958199430801942273 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Let E(0) = x + 1, let E(n+1) = 1 - E(n) + E(n)^2. Let e(n) = discrim(E(n),x) and let f(n) = e(n+1)/e(n)^2. Then f(1,2,3,...) = -3,13,217,57073,381195849,... which looks like this sequence (I do not have a proof yet). - Daniel R. L. Brown (dbrown(AT)certicom.com), Nov 18 2005

This sequence gives the next number in a sequence where the sum and the product of the terms of the sequence are equal.

It happens that the sum or product of the terms of this sequence match A001146 for the numerator of the sum or product and A076628 for the denominator of the sum or product of the sequence.

LINKS

Table of n, a(n) for n=1..10.

N. MacKinnon and N. Lord, Sums equal to products, The Mathematical Gazette, March 1986, 21-22.

Crux Mathematicorum, Mathematical mayhem pb no. 114, Vol 30, 2004, p. 467-468. [Robert FERREOL, Jul 06 2015]

FORMULA

Let F(n) = Product_{i=1..n} f(i) = p/q (say). Then f(n+1) = p/(p-q).

From Robert FERREOL, Jun 12 2015: (Start)

Recurrence: f(1) = f(2) = 2; f(n+1) = f(n)^2/(f(n)^2 - f(n) + 1).

Since f(n) = 2^(2^(n-2))/a(n) for n >= 2, the recurrence for a(n) is:

a(1) = a(2) = 1; a(n+1) = 2^(2^(n-1)) - 2^(2^(n-2))*a(n) + a(n)^2.

(End)

EXAMPLE

2, 2, 4/3, 16/13, 256/217, 65536/57073, 4294967296/3811958497, 18446744073709551616/16605534578235736513, ... = A001146/A100441 (essentially).

MAPLE

f:=proc(n) option remember; local i, k, k1, k2; if n = 1 then return(2); fi; k:=mul(f(i), i=1..n-1); k1:=numer(k); k2:=denom(k); k1/(k1-k2); end;

f:=n-> if n=1 or n=2 then 2 else f(n-1)^2/(f(n-1)^2-f(n-1)+1) fi; # Robert FERREOL, Jun 12 2015

MATHEMATICA

f[n_] := f[n] = (frac = Product[f[i], {i, 1, n-1}]; p = Numerator[frac]; q = Denominator[frac]; p/(p-q)); f[1] = 2; (* or, after Robert FERREOL *) f[n_] := f[n] = If[n == 1 || n == 2, 2, f[n-1]^2/(f[n-1]^2-f[n-1]+1)]; Table[f[n], {n, 1, 10}] // Denominator (* Jean-Fran├žois Alcover, Sep 19 2012, updated Jun 15 2015 *)

PROG

(MAGMA) I:=[1, 3]; [1] cat  [n le 2 select I[n] else 2^(2^(n-1))-2^(2^(n-2))*Self(n-1)+Self(n-1)^2: n in [1..10]]; // Vincenzo Librandi, Jun 13 2015

(PARI) {a(n) = my(s, t); if( n<3, n>0, t = a(n-1); s = 2^(2^(n-3)); s*s -s*t +t*t)}; /* Michael Somos, Aug 05 2017 */

CROSSREFS

Cf. A001146, A076628.

Sequence in context: A085010 A259988 A165903 * A036680 A111431 A015701

Adjacent sequences:  A100438 A100439 A100440 * A100442 A100443 A100444

KEYWORD

nonn,frac,nice

AUTHOR

Gilbert Boily (sgbl(AT)escape.ca), Nov 21 2004, Sep 03 2007

EXTENSIONS

Name edited by Michael Somos, Aug 05 2017

STATUS

approved

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Last modified June 16 03:44 EDT 2019. Contains 324145 sequences. (Running on oeis4.)