A296063 a(n) is the smallest number (in absolute value) not yet in the sequence such that the arithmetic mean of the first n terms a(1), a(2), ..., a(n) is an integer. Preference is given to positive values of a(n); a(1)=1; 0 not allowed.

1, -1, 3, -3, 5, -5, 7, -7, 9, -9, 11, -11, 13, -13, 15, -15, 17, -17, 19, -19, 21, -21, 23, -23, 25, -25, 27, -27, 29, -29, 31, -31, 33, -33, 35, -35, 37, -37, 39, -39, 41, -41, 43, -43, 45, -45, 47, -47, 49, -49, 51, -51

Offset: 1

Author

Enrique Navarrete, Dec 04 2017

Keywords

• sign

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Crossrefs

Cf. A193356 (partial sums), A059841 (a(n)/n), A109613.

Programs

Mathematica

Array[(2 Floor[(# + 1)/2] - 1) (2 Boole@ OddQ@ # - 1) &, 52] (* or *)
Nest[Append[#, Block[{k = 1, s = 1}, While[Nand[FreeQ[#, s k], IntegerQ@ Mean[Append[#, s k]]], If[s == 1, s = -1, k++; s = 1]]; s k]] &, {1}, 51] (* Michael De Vlieger, Dec 12 2017 *)

PARI

Vec(x*(1 + x^2) / ((1 - x)*(1 + x)^2) + O(x^50)) \\ Colin Barker, Mar 14 2020

Formula

a(n) = (-1)^(n+1)*A109613(n+1). - Michel Marcus, Dec 05 2017

From Colin Barker, Mar 14 2020: (Start)

G.f.: x*(1 + x^2) / ((1 - x)*(1 + x)^2).

a(n) = -a(n-1) + a(n-2) + a(n-3) for n>3.

(End)