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A329774 a(n) = n+1 for n <= 2; otherwise a(n) = 3*a(n-3)+1.

%I A329774 #19 Dec 06 2019 13:39:30
%S A329774 1,2,3,4,7,10,13,22,31,40,67,94,121,202,283,364,607,850,1093,1822,
%T A329774 2551,3280,5467,7654,9841,16402,22963,29524,49207,68890,88573,147622,
%U A329774 206671,265720,442867,620014,797161,1328602,1860043,2391484,3985807
%N A329774 a(n) = n+1 for n <= 2; otherwise a(n) = 3*a(n-3)+1.
%C A329774 _Robert Fathauer_ observed that if the "warp and woof" construction used by Jim Conant in his recursive dissection of a square (see A328078) is applied to a triangle, one obtains the Sierpinski gasket.
%C A329774 The present sequence gives the number of regions after the n-th generation of this dissection of a triangle.
%D A329774 Robert Fathauer, Email to N. J. A. Sloane, Oct 14 2019.
%H A329774 Colin Barker, <a href="/A329774/b329774.txt">Table of n, a(n) for n = 0..1000</a>
%H A329774 N. J. A. Sloane, <a href="/A329774/a329774.pdf">Illustration of initial terms.</a>
%H A329774 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3,-3).
%F A329774 From _Colin Barker_, Nov 27 2019: (Start)
%F A329774 G.f.: (1 + x + x^2 - 2*x^3) / ((1 - x)*(1 - 3*x^3)).
%F A329774 a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) for n>3.
%F A329774 (End)
%p A329774 f:=proc(n) option remember;
%p A329774 if n<=2 then n+1 else 3*f(n-3)+1; fi; end;
%p A329774 [seq(f(n),n=0..50)];
%o A329774 (PARI) Vec((1 + x + x^2 - 2*x^3) / ((1 - x)*(1 - 3*x^3)) + O(x^40)) \\ _Colin Barker_, Nov 27 2019
%Y A329774 A mixture of A003462, A060816, and A237930. Cf. A328078.
%K A329774 nonn,easy
%O A329774 0,2
%A A329774 _N. J. A. Sloane_, Nov 27 2019