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A339167 Rounded value of 1/(x[n] + n), where x[n] is the n-th negative solution to Gamma(x) = x^2. 2
-2, 5, -56, 382, -3002, 25918, -246962, 2580478, -29393282, 362879997, -4829932803, 68976230397, -1052366515203, 17086945075197, -294226732800003, 5356234211327997, -102793666719744003, 2074369080655871997, -43913881247588352003, 973160803270655999997 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
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
Sequence in context: A006525 A254406 A260654 * A042161 A176142 A101151
KEYWORD
sign
AUTHOR
M. F. Hasler, Nov 25 2020.
STATUS
approved

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Last modified July 25 01:41 EDT 2024. Contains 374585 sequences. (Running on oeis4.)