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A096531
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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.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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 (noe(AT)sspectra.com), Aug 15 2004
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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 (noe(AT)sspectra.com), 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).
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CROSSREFS
| Cf. A096260, A096227.
Sequence in context: A048757 A173659 A054433 * A149121 A149122 A149123
Adjacent sequences: A096528 A096529 A096530 * A096532 A096533 A096534
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KEYWORD
| nonn
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AUTHOR
| Pierre CAMI (pierre-cami(AT)bbox.fr), Aug 13 2004
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EXTENSIONS
| Corrected and extended by T. D. Noe (noe(AT)sspectra.com), Aug 15 2004
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