A111500 - OEIS

A111500

Number of squares in an n X n grid of squares with diagonals.

1, 10, 31, 72, 137, 234, 367, 544, 769, 1050, 1391, 1800, 2281, 2842, 3487, 4224, 5057, 5994, 7039, 8200, 9481, 10890, 12431, 14112, 15937, 17914, 20047, 22344, 24809, 27450, 30271, 33280, 36481, 39882, 43487, 47304, 51337, 55594, 60079, 64800, 69761, 74970

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OFFSET

0,2

COMMENTS

This sequence is the sum of the number of squares with horizontal/vertical sides (whose length is a positive integer), which is equal to Sum_{j=1..n} j^2 = (n*(n + 1)*(2*n + 1))/6, and the number of squares with diagonal sides (whose length is a multiple of sqrt(2)/2), which is Sum_{j=1..n} (A111746(n - 1)) = floor((n*(4*n^2 - 1))/6). - Marco Ripà, Jan 14 2024

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

FORMULA

a(n) = n^3 + n^2/2 - 1/4 + (1/4)*(-1)^n.

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n > 4. - Colin Barker, May 28 2015

G.f.: (x^3 + 3*x^2 + 7*x + 1) / ((x-1)^4*(x+1)). - Colin Barker, May 28 2015

From Marco Ripà, Jan 14 2024: (Start)

a(n) = Sum_{j=1..n} (A000290(j) + A111746(j-1)).

a(n) = floor(n^3 + n^2/2). (End)

MAPLE

seq(n^3+n^2/2-1/4+1/4*(-1)^n, n=1..65);