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.

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

Links

Table of n, a(n) for n=0..26.

Index entries for linear recurrences with constant coefficients, signature (0,30,0,-280,0,960,0,-1024).

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

Adjacent sequences: A096257 A096258 A096259 * A096261 A096262 A096263

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