A111746 -id:A111746 - cp's OEIS frontend

A330805 Number of squares and rectangles in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.

0, 9, 51, 166, 410, 855, 1589, 2716, 4356, 6645, 9735, 13794, 19006, 25571, 33705, 43640, 55624, 69921, 86811, 106590, 129570, 156079, 186461, 221076, 260300, 304525, 354159, 409626, 471366, 539835, 615505, 698864, 790416, 890681, 1000195, 1119510, 1249194, 1389831

Offset: 0

Luce ETIENNE, Jan 01 2020

Collection: 2*n*(n+1)-ominoes.
Number of squares (all sizes): (8*n^3 + 24*n^2 + 22*n - 3*(-1)^n + 3)/12.
Number of rectangles (all sizes): (8*n^4 + 24*n^3 + 22*n^2 + 3*(-1)^n - 3)/12.

			a(1) = 4*1+5 = 9; a(2) = 4*5+31 = 51; a(3) = 4*15 + 106 = 166; a(4) = 4*36 + 270 = 410.

Teofil Bogdan and Mircea Dan Rus, Counting the lattice rectangles inside Aztec diamonds and square biscuits, arXiv:2007.13472 [math.CO], 2020.
Luce ETIENNE, Illustration of a(1), a(2) and a(3).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A004320, A046092, A111746, A212523.

LinearRecurrence[{5,-10,10,-5,1},{0,9,51,166,410},40] (* Harvey P. Dale, Jun 27 2020 *)

G.f.: x*(x + 3)^2/(1 - x)^5.
E.g.f.: (1/6)*exp(x)*x*(54 + 99*x + 40*x^2 + 4*x^3). - Stefano Spezia, Jan 01 2020
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = n*(n + 1)*(4*n^2 + 12*n + 11)/6.
a(n) = 4*A000332(n+3) + A212523(n+1).
a(n) = 9*A000332(n+3) + 6*A000332(n+2) + A000332(n+1). - Mircea Dan Rus, Aug 26 2020
a(n) = 3*A004320(n) + A004320(n-1). - Mircea Dan Rus, Aug 26 2020

A258440 Number of squares of all sizes in 3n(n+1)/2-ominoes in form of three-quarters of Aztec diamonds.

Original entry on oeis.org

3, 11, 25, 49, 84, 132, 196, 278, 379, 503, 651, 825, 1028, 1262, 1528, 1830, 2169, 2547, 2967, 3431, 3940, 4498, 5106, 5766, 6481, 7253, 8083, 8975, 9930, 10950, 12038, 13196, 14425, 15729, 17109, 18567, 20106, 21728, 23434, 25228, 27111, 29085, 31153, 33317, 35578, 37940, 40404, 42972, 45647, 48431

Offset: 1

Luce ETIENNE, May 30 2015

These polyominoes are 6*n-gons, and thus their number of vertices is n*(3*n+7).

Schäfli's notation for figure corresponding to a(1): 4.4.4.

			a(1)=3, a(2)=9+2=11, a(3)=18+7=25, a(4)=30+15+4=49, a(5)=45+26+11+2=84.

Colin Barker, Table of n, a(n) for n = 1..1000
Luce ETIENNE, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Cf. A000217, A045943, A111746, A140090, A173196, A241526.

[(52*n^3+186*n^2+212*n-3*(32*Floor(n/3)+3*(1-(-1)^n)))/144: n in [1..50]]; // Vincenzo Librandi, Jun 02 2015

A258440:=n->(52*n^3+186*n^2+212*n-3*(32*floor(n/3)+3*(1-(-1)^n)))/144: seq(A258440(n), n=1..100); # Wesley Ivan Hurt, Jul 10 2015

Table[(52 n^3 + 186 n^2 + 212 n - 3 (32 Floor[n/3] + 3 (1 - (-1)^n)))/144, {n, 45}] (* Vincenzo Librandi, Jun 02 2015 *)

Vec(x*(2*x^3+3*x^2+5*x+3)/((x-1)^4*(x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Jun 01 2015

a(n) = 2*A241526(n) - A173196(n+1).
a(n) = (1/8)*(Sum_{i=0..(n-1-floor(n/3)}(4*n+1-6*i-(-1)^i)*(4*n+3-6*i+(-1)^i)- Sum_{j=0..(2*n-1+(-1)^n)}(2*n+1+(-1)^n-4*j)*(2*n+1-(-1)^n-4*j)).
a(n) = (52*n^3+186*n^2+212*n-3*(32*floor(n/3)+3*(1-(-1)^n)))/144.
a(n) = 2*a(n-1)-a(n-3)-a(n-4)+2*a(n-6)-a(n-7) for n>7. - Colin Barker, Jun 01 2015
G.f.: x*(2*x^3+3*x^2+5*x+3) / ((x-1)^4*(x+1)*(x^2+x+1)). - Colin Barker, Jun 01 2015

Typo in data fixed by Colin Barker, Jun 01 2015

Name edited by Michel Marcus, Dec 22 2020

A111500 Number of squares in an n X n grid of squares with diagonals.

Original entry on oeis.org

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

Offset: 0

Floor van Lamoen, Nov 16 2005

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

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).

Cf. A000290, A100583, A111746.

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

Vec((x^3+3*x^2+7*x+1) / ((x-1)^4*(x+1)) + O(x^100)) \\ Colin Barker, May 28 2015

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)

cp's OEIS Frontend

A330805 Number of squares and rectangles in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A258440 Number of squares of all sizes in 3n(n+1)/2-ominoes in form of three-quarters of Aztec diamonds.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A111500 Number of squares in an n X n grid of squares with diagonals.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

cp's OEIS Frontend

A330805 Number of squares and rectangles in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A258440 Number of squares of all sizes in 3*n*(n+1)/2-ominoes in form of three-quarters of Aztec diamonds.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A111500 Number of squares in an n X n grid of squares with diagonals.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

A258440 Number of squares of all sizes in 3n(n+1)/2-ominoes in form of three-quarters of Aztec diamonds.