%I #48 Sep 08 2022 08:46:12

%S 3,11,25,49,84,132,196,278,379,503,651,825,1028,1262,1528,1830,2169,

%T 2547,2967,3431,3940,4498,5106,5766,6481,7253,8083,8975,9930,10950,

%U 12038,13196,14425,15729,17109,18567,20106,21728,23434,25228,27111,29085,31153,33317,35578,37940,40404,42972,45647,48431

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

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

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

%H Colin Barker, <a href="/A258440/b258440.txt">Table of n, a(n) for n = 1..1000</a>

%H Luce ETIENNE, <a href="/A258440/a258440.pdf">Illustration of initial terms</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,2,-1).

%F a(n) = 2*A241526(n) - A173196(n+1).

%F 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)).

%F a(n) = (52*n^3+186*n^2+212*n-3*(32*floor(n/3)+3*(1-(-1)^n)))/144.

%F 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

%F 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

%e 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.

%p 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

%t 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 *)

%o (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

%o (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

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

%K nonn,easy

%O 1,1

%A _Luce ETIENNE_, May 30 2015

%E Typo in data fixed by _Colin Barker_, Jun 01 2015

%E Name edited by _Michel Marcus_, Dec 22 2020