A291196 Third differences of {Pi*n^2}, fractional part of Pi*n^2.

1, 0, -1, 2, -3, 3, -2, 1, 0, 0, -1, 1, -1, 2, -2, 0, 2, -2, 1, 0, -1, 1, 0, -1, 1, 0, 0, -1, 2, -3, 3, -2, 1, 0, -1, 1, 0, -1, 2, -2, 0, 2, -2, 1, -1, 2, -2, 0, 2, -2, 1, -1, 1, 0, 0, 0, -1, 1, -1, 2, -2, 0, 2, -2, 1, -1, 2, -2, 0, 2, -2, 1, 0, -1, 1, 0, -1, 2, -3, 3, -2, 1, 0, 0, -1, 1, 0, -1

Offset

1,4

Comments

From Robert Israel, Aug 20 2017: (Start)

-3 <= a(n) <= 3.

It appears that:

a(n) = -3 if and only if a(n+1) = 3.

if a(n) = -2 then a(n+1) = 0, 1 or 2.

if a(n) = -1 or 0 then a(n+1) = -1, 0, 1 or 2.

if a(n) = 1 then a(n+1) = -3, -1, 0 or 1.

if a(n) = 2 then a(n+1) = -3 or -2.

if a(n) = 3 then a(n+1) = -2 or -1.

(End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = {Pi*(n+3)^2} - 3*{Pi*(n+2)^2} + 3*{Pi*(n+1)^2} - {Pi*n^2}.

Maple

A0:= [seq(frac(Pi*n^2), n=1..103)]:
A1:= A0[2..-1]-A0[1..-2]:
A2:= A1[2..-1]-A1[1..-2]:
A2[2..-1]-A2[1..-2]; # Robert Israel, Aug 20 2017

Mathematica

Differences[FractionalPart[Pi*Range[100]^2], 3] (* Paolo Xausa, Feb 23 2024 *)

Crossrefs

Sequence in context: A167544 A334484 A334234 * A279316 A074989 A307429

Adjacent sequences: A291193 A291194 A291195 * A291197 A291198 A291199

Keyword

sign

Author

Simon Plouffe, Aug 20 2017

Status

approved