A183299 Complement of A005563.

1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88

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

Links

Table of n, a(n) for n=1..80.

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

Cf. A005563.

Sequence in context: A103838 A139283 A188013 * A282532 A039243 A265187

Adjacent sequences: A183296 A183297 A183298 * A183300 A183301 A183302

Keyword

nonn

Author

Clark Kimberling, Jan 03 2011

Status

approved