OFFSET
1,2
COMMENTS
From Andres Cicuttin, Apr 18 2016: (Start)
Defining the probability for integer value x in a Poisson distribution of integer mean = q > 0 as P(q,x) = e^(-q)*(q^x)/x! then it is conjectured that P(n, a(n)+1)- P(n, a(n)) <= P(n, k+1)- P(n, k), for every k > 0.
That is, a(n) is the position where the Poisson distribution with mean n has its minimum discrete difference (not proved, but tested up to n = 20*10^3).
(Very qualitative) Plot of a Poisson Distribution with mean q = n. The vertical line above a(n) indicates the place where the distribution has its minimum (negative) discrete difference.
P
^
| *
| * *
| * *
| * *
| * *
| * | *
| * | *
| * | *
| * | *
*-------------+----+----------------------> x
n a(n)
For instance, if n = 8 then
P(8, a(8)+1) - P(8, a(8)) = P(8,11) - P(8,10) = -0.027071
If we now calculate the discrete difference in a(n)+1 we then obtain
P(8,a(8)+2) - P(8,a(8)+1) = P(8,12)- P (8,11) = -0.0240634
and in a(n)-1
P(8,a(8)) - P(8,a(8)-1) = P(8,10) - P(8,9) = -0.0248154
Both previous values are larger than the minimum obtained at a(n). (End)
Numbers k such that sqrt(k+1) is not an integer. - Wesley Ivan Hurt, Feb 03 2022
FORMULA
(See the Mathematica code.)
MATHEMATICA
a=1; b=2;
F[n_]:=a*n^2+b*n;
R[n_]:=(n/a+((b-1)/(2a))^2)^(1/2);
G[n_]:=n-1+Ceiling[R[n]-(b-1)/(2a)];
Table[F[n], {n, 60}]
Table[G[n], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 03 2011
STATUS
approved