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

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

1,1

Comments

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.

Links

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).

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(PARI) 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
(Magma) [(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

Crossrefs

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

Sequence in context: A320035 A147382 A212971 * A184634 A352013 A164303

Adjacent sequences: A258437 A258438 A258439 * A258441 A258442 A258443

Keyword

nonn,easy

Author

Luce ETIENNE, May 30 2015

Extensions

Typo in data fixed by Colin Barker, Jun 01 2015

Name edited by Michel Marcus, Dec 22 2020

Status

approved