This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339204 #51 Dec 22 2024 13:47:10
%S A339204 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,
%T A339204 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,
%U A339204 7,4,6,4,4,6,6,8,3,4,6,7,6,9,3,0,4,1,7,5
%N A339204 Decimal expansion of the generating constant for the Fibonacci numbers.
%C A339204 Inspired by the prime generating constant A249270, but here for the Fibonacci numbers, A000045(n); generating the Fibonacci numbers for n > 2.
%C A339204 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.
%C A339204 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.
%H A339204 Dylan Friedman, Juli Garbulsky, Bruno Glecer, James Grime, and Massi Tron Florentin, <a href="https://www.researchgate.net/publication/330746181_A_Prime-Representing_Constant">A Prime-Representing Constant</a>, 2019.
%F A339204 Equals Sum_{n > 2} (A000045(n)-1)/(Product_{k = 2..n-1} A000045(k)).
%e A339204 2.95693889137798804983169009791120927869915823439362...
%p A339204 with(combinat, fibonacci): evalf(Sum((fibonacci(n) - 1)/Product(fibonacci(k), k = 2..n-1), n = 3..infinity), 120); # _Vaclav Kotesovec_, Nov 29 2020
%t A339204 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 *)
%o A339204 (PARI) suminf(n=3, (fibonacci(n)-1)/prod(k=2, n-1, fibonacci(k))) \\ _Michel Marcus_, Nov 27 2020
%o A339204 (Python)
%o A339204 n, sumn, sumd, termd, f0, f1 = 0, 0, 1, 1, 1, 1
%o A339204 while n < 33: # enough to obtain 100 digits
%o A339204 n, sumn, sumd, termd, f0, f1 = n+1, sumn*termd+sumd*(f0-1), sumd*termd, termd*f0, f0+f1, f0
%o A339204 pre, sumn, i, d = sumn//sumd, sumn%sumd, 0, ""
%o A339204 while i < 100:
%o A339204 dig, sumn, i = (10*sumn)//sumd, (10*sumn)%sumd, i+1
%o A339204 d = d+str(dig)
%o A339204 print(str(pre)+"."+d)
%Y A339204 Cf. A000045 (Fibonacci).
%Y A339204 Cf. A249270 (for primes), A339203 (for Mersenne prime exponents).
%K A339204 nonn,cons
%O A339204 1,1
%A A339204 _A.H.M. Smeets_, Nov 27 2020