OFFSET
1,1
COMMENTS
Inspired by the prime generating constant A249270, but here for the Fibonacci numbers, A000045(n); generating the Fibonacci numbers for n > 2.
The producing function is given by f' = floor(f)*(f-floor(f)+1), starting with this constant, f' denoting the next f, and floor(f) being the terms of the sequence produced by this constant.
The number of correct digits obtained from the first n terms from the series expansion for this constant as given in the formula section is roughly about (n^2)/10 (~ (3/7)*(log(Fib(n))^2) decimal digits; i.e., for a binary representation, about (n^2)/3 binary digits.
LINKS
Dylan Friedman, Juli Garbulsky, Bruno Glecer, James Grime, and Massi Tron Florentin, A Prime-Representing Constant, 2019.
EXAMPLE
2.95693889137798804983169009791120927869915823439362...
MAPLE
with(combinat, fibonacci): evalf(Sum((fibonacci(n) - 1)/Product(fibonacci(k), k = 2..n-1), n = 3..infinity), 120); # Vaclav Kotesovec, Nov 29 2020
MATHEMATICA
Quiet[First[RealDigits[NSum[(Fibonacci[n] - 1)/Fibonorial[n - 1], {n, 3, Infinity}, Method -> {"WynnEpsilon", "ExtraTerms" -> 25}, NSumTerms -> 25, VerifyConvergence -> False, WorkingPrecision -> 105], 10, 100]], General::intnm] (* Jan Mangaldan, Nov 29 2020 *)
PROG
(PARI) suminf(n=3, (fibonacci(n)-1)/prod(k=2, n-1, fibonacci(k))) \\ Michel Marcus, Nov 27 2020
(Python)
n, sumn, sumd, termd, f0, f1 = 0, 0, 1, 1, 1, 1
while n < 33: # enough to obtain 100 digits
n, sumn, sumd, termd, f0, f1 = n+1, sumn*termd+sumd*(f0-1), sumd*termd, termd*f0, f0+f1, f0
pre, sumn, i, d = sumn//sumd, sumn%sumd, 0, ""
while i < 100:
dig, sumn, i = (10*sumn)//sumd, (10*sumn)%sumd, i+1
d = d+str(dig)
print(str(pre)+"."+d)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
A.H.M. Smeets, Nov 27 2020
STATUS
approved