A235526 - OEIS

%I #6 Jan 23 2014 19:49:48

%S 1,1,2,5,5,6,6,3,2,17,11,10,13,10,12,12,6,6,18,9,6,9,6,6,6,3,2,53,29,

%T 22,31,22,24,24,12,10,37,28,32,32,23,24,24,12,12,30,18,18,18,12,12,12,

%U 6,6,54,27,18,27,18,18,18,9,6,27,18,18,18,12,12,12,6,6,18,9,6,9,6,6,6,3,2,161

%N G.f. satisfies: A(x) = Sum_{n>=0} [A(x)^n (mod 3)]*x^n.

%H Paul D. Hanna, <a href="/A235526/b235526.txt">Table of n, a(n) for n = 0..200</a>

%F a(3^n) = 2*3^n - 1 for n>=0.

%F a(2*3^n) = 2*3^n for n>=0.

%F a(3^n-1) = 2 for n>=1.

%F a(3^n+1) = 3^n + 2 for n>=1.

%F a(n) == binomial(2*n,n)/(n+1) (mod 3); i.e., a(n) is congruent to Catalan number A000108(n) modulo 3.

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 3*x^7 + 2*x^8 + 17*x^9 + 11*x^10 + 10*x^11 + 13*x^12 +...

%e The table of coefficients in A(x)^n reduced modulo 3 begins:

%e n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=1: [1, 1, 2, 2, 2, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, ...];

%e n=2: [1, 2, 2, 2, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, ...];

%e n=3: [1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, ...];

%e n=4: [1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=5: [1, 2, 2, 0, 2, 2, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=6: [1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=7: [1, 1, 2, 1, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=8: [1, 2, 2, 1, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=9: [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, ...];

%e n=10:[1, 1, 2, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, ...];

%e n=11:[1, 2, 2, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, ...];

%e n=12:[1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, ...];

%e n=13:[1, 1, 2, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 1, ...];

%e n=14:[1, 2, 2, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 1, ...];

%e n=15:[1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, ...];

%e n=16:[1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 1, ...];

%e n=17:[1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 1, ...];

%e n=18:[1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, ...];

%e n=19:[1, 1, 2, 2, 2, 0, 0, 0, 2, 1, 1, 2, 2, 2, 0, 0, 0, 1, 0, ...];

%e n=20:[1, 2, 2, 2, 0, 0, 0, 2, 2, 1, 2, 2, 2, 0, 0, 0, 1, 1, 0, ...];

%e n=21:[1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, ...];

%e n=22:[1, 1, 2, 0, 0, 2, 1, 1, 0, 1, 1, 2, 0, 0, 1, 2, 2, 0, 0, ...];

%e n=23:[1, 2, 2, 0, 2, 2, 1, 0, 0, 1, 2, 2, 0, 1, 1, 2, 0, 0, 0, ...];

%e n=24:[1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, ...];

%e n=25:[1, 1, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, ...];

%e n=26:[1, 2, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0],...];

%e n=27:[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],...]; ...

%e where the antidiagonal sums form this sequence.

%o (PARI) {MOD(F,n)=local(V=Vec(F));sum(k=0,#V-1,(V[k+1]%n)*x^k)+O(x^#V)}

%o {a(n)=local(A=1+x);for(i=1,n,A=1+sum(k=1,n,x^k*MOD((A+x*O(x^n))^k,3)));polcoeff(A,n)}

%o for(n=0,40,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 23 2014