|
|
A005436
|
|
Number of convex polygons of perimeter 2n on square lattice.
(Formerly M1778)
|
|
11
|
|
|
1, 2, 7, 28, 120, 528, 2344, 10416, 46160, 203680, 894312, 3907056, 16986352, 73512288, 316786960, 1359763168, 5815457184, 24788842304, 105340982248, 446389242480, 1886695382192, 7955156287456, 33468262290096, 140516110684832, 588832418973280, 2463133441338048
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,2
|
|
COMMENTS
|
Or, a(n) = number of convex polyominoes of perimeter 2n. - David Callan, Jul 25 2008
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (2*n + 3)*4^(n-4) - 4*(n-3)*C(2*n-7, n-4) for n >= 4. - Corrected by Robert Israel, Apr 04 2016
a(n) = (2*n+11)*4^n - 4*(2*n+1)*binomial(2*n,n) produces the terms (except the first two) with a different offset. - N. J. A. Sloane, Oct 14 2017
G.f.: x^2*(1-6*x+11*x^2-4*x^3)/(1-4*x)^2 - 4*x^4*(1-4*x)^(-3/2). - Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003
a(n) = (2*(8*n^2 -106*n +303)*a(n-1) - 8*(2*n-9)*(2*n-17)*a(n-2))/((n-4)*(2*n- 19)), with a(2) = 1, a(3) = 2, a(4) = 7, a(4) = 28. - G. C. Greubel, Nov 20 2022
|
|
MAPLE
|
t1:=x^2*( (1-6*x+11*x^2-4*x^3)/(1-4*x)^2 - 4*x^2/(1-4*x)^(3/2));
series(t1, x, 40);
|
|
MATHEMATICA
|
Join[{1, 2}, Table[(2 n + 11) 4^n - 4 (2 n + 1) Binomial[2 n, n], {n, 0, 25}]] (* Vincenzo Librandi, Jun 25 2015 *)
|
|
PROG
|
(Magma) [1, 2] cat [4^n*(2*n+11)-4*(2*n+1)*Binomial(2*n, n): n in [0..25]]; // Vincenzo Librandi, Jun 25 2015
(SageMath)
def A005436(n): return (2*n+3)*4^(n-4) -4*(n-3)*binomial(2*n-7, n-4) + (9/16)*int(n==2) - (1/4)*int(n==3)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|