Muhammed Sefa Saydam has authored 2 sequences.
A387202
a(n) is the number of dissections of a (4*n+2)-gon into hexagons using strictly disjoint diagonals.
Original entry on oeis.org
1, 5, 21, 87, 363, 1534, 6570, 28492, 124944, 553301, 2471373, 11122275, 50389695, 229643895, 1052093655, 4842863465, 22386911925, 103885321615, 483759492255, 2259888333445, 10587902977185, 49738841822400, 234235771140876, 1105609645231322, 5229610939919718
Offset: 1
A380591
a(n) is the number of dissections of a convex (n+2)-sided polygon by nonintersecting diagonals into triangles and quadrilaterals such that at least one of the dividing diagonals passes through a chosen vertex.
Original entry on oeis.org
0, 1, 5, 21, 90, 395, 1773, 8110, 37686, 177450, 844935, 4061762, 19687020, 96107358, 472132330, 2332304055, 11578595554, 57736664825, 289055592810, 1452381167325, 7321620080550, 37020073600755, 187699184460450, 954084756674088, 4861008765722340
Offset: 1
a(2) = 1. Because: Let's choose vertex A in convex quadrilateral ABCD. At least one diagonal must pass through corner A. A diagonal is drawn at corner C. 2 triangles are obtained. There is 1 situation.
a(3) = 5. Because: Let's choose vertex A in convex pentagon ABCDE. At least one diagonal must pass through corner A. First case: diagonals AD and AC can be drawn. Second Case: Diagonals AD and DB can be drawn. Third case: Only diagonal AD can be drawn. Fourth Case: Diagonals AC and EC can be drawn. Fifth Case: Only diagonal AC can be drawn. There are 5 situations in total.
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a:= proc(n) option remember; `if`(n<3, n*(n-1)/2, (n*(3059*n^2
-7876*n+4997)*a(n-1)+(4120*n^3-22681*n^2+39305*n-21644)*a(n-2)
+39*(n-3)*(3*n-10)*(3*n-11)*a(n-3))/(5*(n+1)*n*(142*n-337)))
end:
seq(a(n), n=1..25); # Alois P. Heinz, Jan 27 2025
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a(n) = if(n==0,1,if(n==1,-1,-sum(i=ceil(n/2),n,binomial(i,n-i)*binomial(n+i-2,n-2)/(n-1)))) + (1/(n+1))*sum(k=ceil(n/2),n,binomial(n+k,k)*binomial(k,n-k)); \\ Michel Marcus, Jan 27 2025