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A338996
Numbers of squares and rectangles of all sizes in 3*n*(n+1)/2-ominoes in form of three-quarters of Aztec diamonds.
1
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
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 *)
KEYWORD
nonn,easy
AUTHOR
Luce ETIENNE, Nov 18 2020
STATUS
approved