A096531 Number of different squares created when a square sheet of paper is folded n times, the first time by one of the diagonal of the square and after by the median of the triangle.

0, 0, 4, 9, 34, 71, 245, 543, 1835, 4223, 14167, 33279, 111279, 264191, 882015, 2105343, 7023295, 16809983, 56055167, 134348799, 447916799, 1074266111, 3581236735, 8592031743, 28641504255, 68727865343, 229098477567

Offset

1,3

Comments

There are two types of squares: (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 squares are enumerated by the p(x) and d(x) functions. - T. D. Noe, Aug 15 2004

Links

Table of n, a(n) for n=1..27.

Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-56,56,64,-64).

Formula

a(1)=0, a(2)=0, a(3)=4, a(5)=34, a(6)=71, a(7)=245, a(8)=509 are easily computed. If n even > 8 define Y(8)=130, Y(n)=3*Y(n-2) and then a(n)=9*a(n-2)-3*Y(n-2); if n odd define Y(7)=27, Y(n)=6*Y(n-2)-3 and then a(n)=8*a(n-2)+3-6*Y(n-2)

Let p(x) = x(x+1)(2x+1)/6 and d(x) = x(4x+1)(4x-1)/3. Then, for n>3, a(n) = -1 + p(2^ceiling(n/2-1)) + d(2^floor(n/2-2)) - T. D. Noe, Aug 15 2004

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

G.f.: -x^3*(32*x^7-60*x^5-48*x^4+33*x^3+31*x^2-5*x-4)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(8*x^2-1)). [Colin Barker, Oct 21 2012]

Crossrefs

Cf. A096260, A096227.

Sequence in context: A054433 A219769 A308038 * A149121 A149122 A149123

Adjacent sequences: A096528 A096529 A096530 * A096532 A096533 A096534

Keyword

nonn,easy

Author

Pierre CAMI, Aug 13 2004

Extensions

Corrected and extended by T. D. Noe, Aug 15 2004

Status

approved