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

0,2

Links

Table of n, a(n) for n=0..30.

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).

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).

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A064675 A135713 A085740 * A349919 A212783 A226315

Adjacent sequences: A338993 A338994 A338995 * A338997 A338998 A338999

Keyword

nonn,easy

Author

Luce ETIENNE, Nov 18 2020

Status

approved