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A096260
Number of different triangles, squares and rectangles created when a square piece of paper is folded n times, the first time by a diagonal of the square and after by the median of the triangle.
2
1, 3, 9, 25, 62, 205, 534, 2110, 5844, 25996, 75592, 359256, 1080592, 5314480, 16315424, 81638240, 253481024, 1279358656, 3996074112, 20256075136, 63463817472, 322392513280, 1011648561664, 5144661112320, 16156254536704, 82205698518016, 258259323717632
OFFSET
0,2
COMMENTS
After unfolding the paper, the count consists of (a) triangles whose hypotenuse is parallel to the sides of the original square, (b) triangles whose hypotenuse is diagonal to the sides of the original square, (c) rectangles whose sides are parallel to the sides of the original square and (d) rectangles whose sides are diagonal to the sides of the original square. Note that the count for (c) is A096222(2*(Ceiling(n/2)-1)). E.g. a(5) = 205 because the counts are a=32, b=64, c=100, d=9. - T. D. Noe, Aug 09 2004
FORMULA
a(n) = A096227(n) + A097405(n) + 1
G.f.: -(1024*x^11+512*x^10+832*x^9+432*x^8-80*x^7-234*x^6-295*x^5-72*x^4+65*x^3+21*x^2-3*x-1) / ((2*x-1)*(2*x+1)*(4*x-1)*(4*x+1)*(2*x^2-1)*(8*x^2-1)). [Colin Barker, Nov 23 2012]
MATHEMATICA
Join[{1, 3, 9, 25}, LinearRecurrence[{0, 30, 0, -280, 0, 960, 0, -1024}, {62, 205, 534, 2110, 5844, 25996, 75592, 359256}, 20]] (* Harvey P. Dale, Apr 08 2015 *)
CROSSREFS
Cf. A096222.
Cf. A096227 (triangles created), A097405 (rectangles created).
Sequence in context: A004255 A065971 A145127 * A375135 A292326 A195417
KEYWORD
nonn,easy
AUTHOR
Pierre CAMI, Aug 01 2004
EXTENSIONS
Corrected and extended by T. D. Noe, Aug 09 2004 and Aug 16 2004
More terms from Colin Barker, Nov 23 2012
STATUS
approved