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A096227 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. 3
2, 8, 16, 44, 96, 268, 648, 1832, 4784, 13456, 36832, 102944, 289216, 804928, 2292608, 6365312, 18257664, 50626816, 145731072, 403833344 (list; graph; refs; listen; history; text; internal format)



Table of n, a(n) for n=1..20.

Index entries for linear recurrences with constant coefficients, signature (0,14,0,-56,0,64).


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)

For n>3, satisfies a linear recurrence with characteristic polynomial (1-2x)(1+2x)(1-2x^2)(1-8x^2).

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]


For n odd X(5)=8 Y(5)=2

X(7)=17 Y(7)=3

X(9)=43 Y(9)=5

X(11)=127 Y(11)=9


Sequence in context: A232392 A176143 A296946 * A191309 A323351 A134353

Adjacent sequences:  A096224 A096225 A096226 * A096228 A096229 A096230




Pierre CAMI, Aug 11 2004



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Last modified December 16 04:05 EST 2019. Contains 330013 sequences. (Running on oeis4.)