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.
Let g(x) = x^2 - x + 1 be the map producing Sylvester's sequence A000058. Then for n >= 0, g^n(1/2) = 1/f(n+2), where g^n is the n-th iterate of g, so a(n+2) is the numerator of g^n(1/2). - Curtis Bechtel, Apr 05 2024
LINKS
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
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-François 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 */
(SageMath)
@CachedFunction
def a(n): # a = A100441
if (n<3): return 2*n-1
else: return 2^(2^(n-1)) - 2^(2^(n-2))*a(n-1) + a(n-1)^2
[1]+[a(n) for n in range(1, 12)] # G. C. Greubel, Apr 08 2023
CROSSREFS
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