A328152 a(n) is the number of squares of side length greater than 1 having vertices at the points of an n X n grid of dots.

0, 0, 2, 11, 34, 80, 160, 287, 476, 744, 1110, 1595, 2222, 3016, 4004, 5215, 6680, 8432, 10506, 12939, 15770, 19040, 22792, 27071, 31924, 37400, 43550, 50427, 58086, 66584, 75980, 86335, 97712, 110176, 123794, 138635, 154770, 172272, 191216, 211679, 233740

Offset

1,3

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n^2*(n^2 - 1)/12 - (n - 1)^2.

From Colin Barker, Oct 06 2019: (Start)

G.f.: x^3*(2 - x)*(1 + x) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

a(n) = (-12 + 24*n - 13*n^2 + n^4) / 12.

(End)

E.g.f.: 1 + (1/12)*exp(x)*(-12 + 12*x - 6*x^2 + 6*x^3 + x^4). - Stefano Spezia, Oct 06 2019

Example

On the 4 X 4 grid we have: four 2 X 2 squares, four sqrt(2) X sqrt(2) squares, two sqrt(5) X sqrt(5) squares, and one 3 X 3 square. Hence, a(4) = 11.

Mathematica

Table[(n^4 - n^2)/12 - (n - 1)^2, {n, 1, 41}]

Prog

(PARI) a(n) = n^2*(n^2 - 1)/12 - (n - 1)^2; \\ Michel Marcus, Oct 06 2019
(PARI) concat([0, 0], Vec(x^3*(2 - x)*(1 + x) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Oct 06 2019

Crossrefs

Cf. A002415.

Sequence in context: A056359 A088306 A100109 * A209033 A222650 A026961

Adjacent sequences: A328149 A328150 A328151 * A328153 A328154 A328155

Keyword

nonn,easy

Author

Derek J. Graves, Oct 05 2019, on behalf of his 2019-20 Geometry class

Status

approved