

A243989


Rounded down ratio of a lune area and a unit circle one, the lune is bounded by two unit circles whose centers are separated by a distance 1/n.


1



1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 25, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 43, 45, 47, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 65, 67, 69, 70, 72, 73, 75, 76, 78, 80, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 98, 100, 102, 103
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OFFSET

1,2


COMMENTS

It seems to be for n >= 10 the first difference will be periodic of {1, 2, 1, 2, 1, 2, 2}.


LINKS

Eric Weisstein's World of Mathematics, Lune


FORMULA

a(n) = floor(Pi/area) where area = 2*d + k1  k2, d = (1/4)*(c^2*(2+c)*(2c))^(1/2), k1 = arccos(c^2/(2*c)), k2 = arccos(c^2/(2*c)).
Empirical g.f.: x*(x^14x^13+2*x^6+2*x^5+x^4+2*x^3+x^2+2*x+1) / ((x1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)).  Colin Barker, Jun 17 2014


PROG

(PARI)
{
for (n=1, 100, c=1/n;
d = (1/4)*(c^2*(2+c)*(2c))^(1/2);
k1 = acos(c^2/(2*c));
k2 = acos(c^2/(2*c));
area = 2*d+k1k2;
print1(floor(Pi/area), ", ")
)
}


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



