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FORMULA
| a(0) = 1, a(n) = 9*a(n-1) - 4.
a(n) = sum(0<=k<=n, binomial(2*n, 2*k)*4^k ).
a(n) = A002438(n) / A000364(n); A000364(n) : Euler numbers.
G.f.: (1-5*x)/((1-x)*(1-9*x)).
a(n)=(3^n+1^n+(-1)^n+(-3)^n)/4.
E.g.f.: exp(3*x)+exp(x)+exp(-x)+exp(-3*x).
Each term expresses a Pythagorean relationship, along with (a(n)-1) and a power of 3, n>0, such that sqrt((a(n))^2 - (a(n)-1)^2) = 3^n. E.g. 365^2 - 364^2 - 3^3 = 27. (the Pythagorean triangle (365, 364, 27). - Gary W. Adamson (qntmpkt(AT)yahoogroups.com), Jun 25 2006
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EXAMPLE
| From Adi Dani, May 28 2011: (Start)
a(2)=41: there are 41 compositions of even natural numbers into 2 parts <=8:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(4,8),(8,4),(5,7),(7,5),(6,6);
(6,8),(8,6),(7,7);
(8,8). (End)
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