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%I #82 Apr 28 2022 13:46:12
%S 3,6,15,45,153,561,2145,8385,33153,131841,525825,2100225,8394753,
%T 33566721,134242305,536920065,2147581953,8590131201,34360131585,
%U 137439739905,549757386753,2199026401281,8796099313665,35184384671745
%N The (2^n+1)-th triangular number (cf. A000217).
%C Number of types of Boolean functions of n variables under a certain group.
%C Also the number of ordered decompositions of 2^n into 3 nonnegative integers (e.g., 2 = 0+0+2 = 0+2+0 = 2+0+0 = 1+1+0 = 1+0+1 = 0+1+1). - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007
%H Vincenzo Librandi, <a href="/A028401/b028401.txt">Table of n, a(n) for n = 2..1000</a>
%H Daniel Poveda Parrilla, <a href="/A028401/a028401.gif">Illustration of initial terms</a>
%H I. Strazdins, <a href="http://dx.doi.org/10.1023/A:1005769927571">Universal affine classification of Boolean functions</a>, Acta Applic. Math. 46 (1997), 147-167.
%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F From _Ralf Stephan_, Aug 23 2003: (Start)
%F a(n) = (3/8)*2^n + (1/32)*4^n + 1.
%F a(n) = 3*A007581(n-2) = (3/4)*A060919(n-1). (End)
%F a(n) = (2^n+4)*(2^n+8)/32. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007
%F G.f.: 3*x^2*(1-5*x+5*x^2)/((1-x)*(1-2*x)*(1-4*x)). - _Colin Barker_, Mar 09 2012
%F a(n) = a(n-1) + 3*A000217(2^(n-3)) for n > 2. - _Daniel Poveda Parrilla_, Dec 27 2016
%t Drop[#, 2] &@ CoefficientList[Series[3 x^2*(1 - 5 x + 5 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* _Michael De Vlieger_, Jul 08 2019 *)
%Y Equals 2*A036562(n-4) - 1, n > 3.
%Y Cf. A000217.
%K nonn,easy
%O 2,1
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Feb 24 2000
%E Simpler definition from Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007
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