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A341896 a(n) is the number of words of length n over the alphabet {a,b,c} with an even number of appearances of the letter 'a' and the sum of appearances of the letters 'b' and 'c' add up to at most 3.

%I A341896 #64 Mar 01 2021 18:33:23
%S A341896 1,2,5,14,25,90,61,294,113,690,181,1342,265,2314,365,3670,481,5474,
%T A341896 613,7790,761,10682,925,14214,1105,18450,1301,23454,1513,29290,1741,
%U A341896 36022,1985,43714,2245,52430,2521,62234,2813,73190,3121,85362,3445,98814,3785,113610
%N A341896 a(n) is the number of words of length n over the alphabet {a,b,c} with an even number of appearances of the letter 'a' and the sum of appearances of the letters 'b' and 'c' add up to at most 3.
%D A341896 Rodrigo de Castro, Teoría de la computación, 2004, unilibros.
%H A341896 Luis Mantilla, <a href="/A341896/b341896.txt">Table of n, a(n) for n = 0..46</a>
%H A341896 Luis Mantilla, <a href="/A341896/a341896_2.pdf">demonstration</a>
%H A341896 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F A341896 a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8).
%F A341896 G.f.: (10*x^7-13*x^6+46*x^5+11*x^4+6*x^3+x^2+2*x+1)/((x-1)^4*(x+1)^4).
%F A341896 a(n) = 2*n + 8*C(n,3) if n is odd, a(n) = 1 + 4*C(n,2) if n is even. - _Alois P. Heinz_, Mar 01 2021
%e A341896 a(0) = 1 : the empty word.
%e A341896 a(1) = 2 : {b, c}.
%e A341896 a(2) = 5 : {aa, bb, cc, bc, cb}.
%e A341896 a(3) = 14 : {aab, aac, aba, aca, baa, bbb, bbc, bcb, bcc, caa, cbb, cbc, ccb, bbb}.
%e A341896 a(4) = 25 : {aaaa, aabb, aabc, aacb, aacc, abab, abac, abba, abca, acab, acac, acba, baab, baac, baba, baca, bbaa, bcaa, caab, caac, caba, caca, cbaa, ccaa, acca}.
%Y A341896 Bisection gives: A080856 (even part).
%K A341896 nonn,easy
%O A341896 0,2
%A A341896 _Luis Mantilla_, Feb 28 2021