OFFSET
1,1
COMMENTS
Consider the equation (x-1)! = x^2. The only solution in the positive integers is x = 1, and (5-1)! ~ 5^2 is a near miss.
If (x-1)! is replaced by Gamma(x), then in addition to the positive non-integer solution x = 5.0367... (cf. A264785) there are non-integer solutions increasingly close to each negative integer: x[1] = -1.5259..., x[2] = -1.806544..., x[3] = -3.017901..., x[4] = -3.997382..., x[5] = -5.000333... etc. They are to the left of odd and to the right of even negative integers, and the present sequence gives the reciprocals of the distances 1/(x[n]+n), rounded to nearest integers.
We notice an intriguing pattern in the last digits of the terms, which are of the form m*10^k - 3 (resp. -2 for a(4 .. 9)) for increasingly large k. (Positive terms end in ...97, negative terms end in ...03) Is this a coincidence? Will that pattern prevail? Why these values, off by -3 from multiples of 10^k? We would appreciate a simple explanation of this observation.
EXAMPLE
The largest negative solution to Gamma(x) = x^2 is x[1] = -1.525796..., its (signed) distance from -1 is x[1] + 1 = -0.525796..., the reciprocal is 1.901..., which rounded to nearest integer gives a(1) = 2.
The next negative solution to Gamma(x) = x^2 is x[2] = -1.806544..., its (signed) distance from -2 is x[2] + 2 = +0.193..., with reciprocal 5.169..., and rounded to nearest integer, a(2) = 5.
PROG
(PARI) a(n)={1\/(n-solve(x=n+(-1/(n+2))^(n+1), n-(-2/(n+2))^n, gamma(-x)-x^2))}
CROSSREFS
KEYWORD
sign
AUTHOR
M. F. Hasler, Nov 25 2020.
STATUS
approved