A178789 - OEIS
3, 6, 18, 66, 258, 1026, 4098, 16386, 65538, 262146, 1048578, 4194306, 16777218, 67108866, 268435458, 1073741826, 4294967298, 17179869186, 68719476738, 274877906946, 1099511627778, 4398046511106, 17592186044418, 70368744177666
COMMENTS
Starting from an equilateral triangle, at each step each straight segment is replaced by a "_/\_" shape of four segments of equal length, with the acute angle in the middle pointing to the exterior. The sequence counts the angles which are (i.e., already were) at both extremities, plus the one newly created acute angle in the middle of each former segment. At step n, there are 3*4^(n-1) straight segments, therefore a(n+1) = a(n) + 3*4^(n-1). - M. F. Hasler, Dec 17 2013
FORMULA
G.f.: 3*x*(1 - 3*x)/(1 - 5*x + 4*x^2).
a(n+1) = a(n) + 3*4^(n-1) = a(n) + A002001(n) for n > 0. - M. F. Hasler, Dec 17 2013
MATHEMATICA
a=b=3; lst={a}; Do[a=a+b; b*=4; AppendTo[lst, a], {n, 40}]; lst
Flatten[Table[2^(2*(n-1)) + 2, {n, 1, 50}]](* or *) CoefficientList[Series[(3 - 9*x)/(1 - 5*x + 4*x^2), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 02 2013 *)