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%I #12 Jun 13 2015 00:51:23
%S 2,8,16,44,96,268,648,1832,4784,13456,36832,102944,289216,804928,
%T 2292608,6365312,18257664,50626816,145731072,403833344
%N Number of different triangles created when a square sheet of paper is folded n times, the first time by one of the diagonal of the square sheet and the by the median of the square triangle.
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,14,0,-56,0,64).
%F a(1)=2, a(2)=8, a(3)=16, a(4)=44, a(5)=96 are easily counted. Now if n even > 4 define X(4)=10 and X(n)=2*(X(n-2)-1), then a(n)=3*2^(3*(n/2)-1) + 2^((n/2)-1)*(2*X(n-2)-1); if n odd > 5 define X(5)=8 and Y(5)=2, X(n)=4*X(n-2)-5*(2*Y(n-2)-1) and Y(n)=2*Y(n-2)-1 then a(n)=2^(3*(((n+1)/2)-1)) + 2^(((n+1)/2)-1)*(4*X(n-2)-5*(2*(Y(n-2)-1)
%F For n>3, satisfies a linear recurrence with characteristic polynomial (1-2x)(1+2x)(1-2x^2)(1-8x^2).
%F G.f.: -2*x*(32*x^8+16*x^7+36*x^6+50*x^5-8*x^4-34*x^3-6*x^2+4*x+1)/((2*x-1)*(2*x+1)*(2*x^2-1)*(8*x^2-1)). [_Colin Barker_, Oct 21 2012]
%e For n odd X(5)=8 Y(5)=2
%e X(7)=17 Y(7)=3
%e X(9)=43 Y(9)=5
%e X(11)=127 Y(11)=9
%K nonn,easy
%O 1,1
%A _Pierre CAMI_, Aug 11 2004
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