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