A339204 Decimal expansion of the generating constant for the Fibonacci numbers.

2, 9, 5, 6, 9, 3, 8, 8, 9, 1, 3, 7, 7, 9, 8, 8, 0, 4, 9, 8, 3, 1, 6, 9, 0, 0, 9, 7, 9, 1, 1, 2, 0, 9, 2, 7, 8, 6, 9, 9, 1, 5, 8, 2, 3, 4, 3, 9, 3, 6, 2, 3, 5, 3, 4, 5, 7, 2, 4, 4, 6, 2, 7, 2, 3, 7, 5, 2, 7, 4, 6, 4, 4, 6, 6, 8, 3, 4, 6, 7, 6, 9, 3, 0, 4, 1, 7, 5

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

Table of n, a(n) for n=1..88.

Dylan Friedman, Juli Garbulsky, Bruno Glecer, James Grime, and Massi Tron Florentin, A Prime-Representing Constant, 2019.

Formula

Equals Sum_{n > 2} (A000045(n)-1)/(Product_{k = 2..n-1} A000045(k)).

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

Cf. A000045 (Fibonacci).

Cf. A249270 (for primes), A339203 (for Mersenne prime exponents).

Sequence in context: A201397 A077124 A051491 * A253169 A339925 A171052

Adjacent sequences: A339201 A339202 A339203 * A339205 A339206 A339207

Keyword

nonn,cons

Author

A.H.M. Smeets, Nov 27 2020

Status

approved