A097405 Number of different rectangles created when a square sheet of paper is folded n times, the first time by one of the diagonals of the square and after by the median of the triangle.

0, 0, 8, 17, 108, 265, 1461, 4011, 21211, 62135, 322423, 977647, 5025263, 15510495, 79345631, 247115711, 1261100991, 3945447295, 20110344063, 63059984127, 321227980543, 1008422616575, 5135350103551, 16130465856511, 82131231439871

Offset

1,3

Comments

There are two types of rectangles: (1) those whose edges are parallel to the edges of the initial square and (2) those whose edges are diagonal to the edges of the initial square. These rectangles are enumerated by the p(x) and d(x) functions.

Links

Table of n, a(n) for n=1..25.

Index entries for linear recurrences with constant coefficients, signature (1,30,-30,-280,280,960,-960,-1024,1024).

Formula

Let p(x) = x^2 (x+1)^2/4 and d(x) = (x^4 - x^2 - 6 x)/24. Then, for n>1, a(n) = -1 + p(2^ceiling(n/2-1)) + d(2^floor(n/2))

G.f.: x^3*(256*x^7+880*x^6-360*x^5-706*x^4+113*x^3+149*x^2-9*x-8) / ((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]

Crossrefs

Cf. A096260, A096227, A096531.

Sequence in context: A177129 A177178 A357678 * A192282 A088588 A041537

Adjacent sequences: A097402 A097403 A097404 * A097406 A097407 A097408

Keyword

nonn,easy

Author

T. D. Noe, Aug 16 2004

Status

approved