OFFSET
1,5
LINKS
Mireille Bousquet-Mélou, Convex polyominoes and algebraic languages, Journal of Physics A25 (1992), 1935-1944.
M.-P. Delest and G. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Sci., 34 (1984), 169-206.
Ira M. Gessel, On the number of convex polyominoes, Ann. Sci. Math. Québec 24 (2000), no. 1, 63-66.
K. Y. Lin and S. J. Chang, Rigorous results for the number of convex polygons on the square and honeycomb lattices, Journal of Physics A21 (1988), 2635-2642.
FORMULA
a(w, h) = binomial(2w+2h-4, 2w-2) + ((2w+2h-5)/2)*binomial(2w+2h-6, 2w-3) - 2(w+h-3)*binomial(w+h-2, w-1)*binomial(w+h-4, w-2), for w > 0, h > 0.
a(w, h) = A093118(w-1, h-1).
EXAMPLE
For w=3 and h=2, the a(3,2)=13 polyominoes spanning a 3 X 2 rectangle are
XXX X XX X XX
XXX XXX XX XXX XXX
plus all different horizontal and vertical reflections (1+2+2+4+4=13).
Table begins
1 1 1 1 1 1 1 ...
1 5 13 25 41 61 ...
1 13 68 222 555 ...
1 25 222 1110 ...
1 41 555 ...
1 61 ...
1 ...
MATHEMATICA
Table[Binomial[2 # + 2 h - 4, 2 # - 2] + ((2 # + 2 h - 5)/2) Binomial[2 # + 2 h - 6, 2 # - 3] - 2 (# + h - 3) Binomial[# + h - 2, # - 1] Binomial[# + h - 4, # - 2] &[w - h + 1], {w, 10}, {h, w}] // Flatten (* Michael De Vlieger, Apr 15 2019 *)
PROG
(Sage)
def a(w, h):
s = w+h # half the perimeter
return ( binomial(2*s-4, 2*w-2) + binomial(2*s-6, 2*w-3)*(s-5/2)
- 2*(s-3)*binomial(s-2, w-1)*binomial(s-4, w-2) )
CROSSREFS
KEYWORD
AUTHOR
Günter Rote, Feb 12 2019
STATUS
approved