A179167 Place a(n) red and b(n) blue balls in an urn; draw 3 balls without replacement; Probability(3 red balls) = Probability(1 red and 2 blue balls); binomial(a(n),3) = binomial(a(n),1)*binomial(b(n),2).

3, 4, 11, 37, 134, 496, 1847, 6889, 25706, 95932, 358019, 1336141, 4986542, 18610024, 69453551, 259204177, 967363154, 3610248436, 13473630587, 50284273909, 187663465046, 700369586272, 2613814880039, 9754889933881

Offset: 1

Author

Paul Weisenhorn, Jun 30 2010

Keywords

• nonn
• uned
• easy

Examples

			For n=4, a(4)=37; b(4)=21; binomial(37,3) = 7770;
binomial(37,1)*binomial(21,2) = 37*210 = 7770.
		

Links

• J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
• Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Crossrefs

b(n)=A101265(n).

Programs

Maple

r:=sqrt(3): for n from 1 to 40 do
a(n):=(6+(1+r)*(2+r)^(n-1)+(1-r)*(2-r)^(n-1))/4: end do:

Formula

a(n+2) = 4*a(n+1) - a(n) - 3;

a(n+3) = 5*(a(n+2) - a(n+1)) + a(n); r=sqrt(3);

a(n) = (6 + (1+r)*(2+r)^(n-1) + (1-r)*(2-r)^(n-1))/4;

a(n) = ceiling((6 + (1+r)*(2+r)^(n-1))/4).

From Colin Barker, Dec 11 2012: (Start)

a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n > 4.

G.f.: x*(x^3-6*x^2+11*x-3) / ((x-1)*(x^2-4*x+1)). (End)