%I #13 Jul 09 2021 21:47:39
%S 0,0,0,1,2,4,7,16,32,62,131,264,537,1056,2103,4207,8389,16754,33521,
%T 67037,133943,267788,535388,1070008,2138723,4275407,8544670,17077641
%N a(n) is the number of consecutive even prime gap pairs (g1, g2) satisfying g1 == 0 (mod 6) and g2 == 2 (mod 6) out of the first 2^n consecutive even prime gap pairs.
%C It seems that the fraction of prime gap pairs (g1, g2) for which g1 == 0 (mod 6), satisfying g2 == 2 (mod 6), i.e., a(n)/A340948(n), tends to a constant, say c, when the number of prime gaps tends to infinity. From n = 27 we obtain that c < 0.284, while it can be argued heuristically that c > 0.25.
%C Futhermore, it is believed that a(n) - A345334(n) will change sign infinitely often.
%F a(n) = A340948(n) - (A345332(n) + A345334(n)).
%Y Cf. A340948, A341531, A341532, A345332, A345334.
%K nonn,more
%O 0,5
%A _A.H.M. Smeets_, Jun 14 2021