A305559 [0, -1, -1] together with A000290.

0, -1, -1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

Offset: 0

Author

Paul Curtz, Jun 21 2018

Keywords

• sign
• easy
• less

Comments

Squares leading to an autosequence of the first kind.

The third sequence of the array

A060576(n+1)= 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

A289207(n)= 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, ...

a(n)= 0, -1, -1, 0, 1, 4, 9, 16, 25, 36, ...

0, 10, 10, 5, 0, 1, 8, 27, 64, 125, ...

0, -113, -113, -68, -23, 0, 1, 16, 81, 256, ... .

The first full vertical is (-1)^n*A033312(n).

From 0, the first two nonzero antidiagonals are 0, -1, 10, -113, 1526, ... = (-1)^n* A176824(n+1).

See OEIS Wiki, Autosequence.

a(n) difference table:

0, -1, -1, 0, 1, 4, 9, 16, 25, ...

-1, 0, 1, 1, 3, 5, 7, 9, 11, ...

1, 1, 0, 2, 2, 2, 2, 2, 2, ...

0, -1, 2, 0, 0, 0, 0, 0, 0, ...

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A003992, A033312, A060576, A176824, A289207, A000290, A113679.

Programs

Mathematica

Join[{0,-1,-1},Range[0,100]^2] (* Paolo Xausa, Nov 13 2023 *)

Formula

From Stefano Spezia, May 28 2025: (Start)

G.f.: x*(1 - 2*x + x^3 - 2*x^4)/(1 - x)^3.

E.g.f.: 9 + 5*x + x^2 - exp(x)*(9 - 5*x + x^2). (End)