A338996 Numbers of squares and rectangles of all sizes in 3*n*(n+1)/2-ominoes in form of three-quarters of Aztec diamonds.

0, 5, 27, 85, 205, 420, 770, 1302, 2070, 3135, 4565, 6435, 8827, 11830, 15540, 20060, 25500, 31977, 39615, 48545, 58905, 70840, 84502, 100050, 117650, 137475, 159705, 184527, 212135, 242730, 276520

Offset: 0

Author

Luce ETIENNE, Nov 18 2020

Keywords

• nonn
• easy

Examples

			a(1) = 2*3-1 = 5, a(2) = 2*16-5 = 27, a(3) = 2*50-15 = 85, a(4) = 2*120-35 = 205, a(5) = 2*245-70 = 420, a(6) = 2*448-126 = 770.
		

Links

• Luce ETIENNE, Illustration of a(1), a(2), a(3) and a(4)
• Wikipedia, Aztec diamond.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A000332, A002417, A004320, A045943, A258440, A330805.

Programs

Mathematica

CoefficientList[Series[x (2 x + 5)/(1 - x)^5, {x, 0, 30}], x] (* Michael De Vlieger, Dec 12 2020 *)

Formula

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

E.g.f.: exp(x)*x*(120 + 204*x + 76*x^2 + 7*x^3)/24. - Stefano Spezia, Nov 18 2020

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

a(n) = n*(n + 1)*(n + 2)*(7*n + 13)/24.

a(n) = 2*A004320(n) - A000332(n+3).

a(n) = 2*A000332(n+2) + 5*A000332(n+3).