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A096222
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Number of different rectangles when a piece of paper is folded n times in alternate directions.
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1
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1, 3, 9, 30, 100, 360, 1296, 4896, 18496, 71808, 278784, 1098240, 4326400, 17172480, 68161536, 271589376, 1082146816, 4320165888, 17247043584, 68920934400, 275415040000, 1101122764800, 4402342526976, 17605073043456, 70403108110336
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Similar to A000537, which counts all possible rectangles in an n X n array of squares. In this sequence we count the rectangles in an a X b array of squares, where a=2^floor(n/2) and b=2^ceiling(n/2). Note that a(n) is the product of two triangular numbers.
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FORMULA
| a(n) = a b (a+1) (b+1)/4, where a=2^floor(n/2) and b=2^ceiling(n/2).
a(n) (mod 10^k) is cyclic. For (mod 10) the cycle is 0, 0, 0, 6, 6, 6, 8, 4. - Robert G. Wilson v Jul 31 2004. - Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 31 2004
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EXAMPLE
| a(1) = 3: fold a 1 X 2 rectangle down the middle; there are 3 rectangles, the one on the left, the one on the right and the one we started with. a(2) = 9 : fold a 2 X 2 recatngle along the X and Y axes; there 4 rectangles of size 1 X 1, 4 of size 1 X 2 or 2 X 1 and 1 of size 2 X 2.
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MATHEMATICA
| Table[a=2^Floor[n/2]; b=2^Ceiling[n/2]; Sum[i*j, {i, a}, {j, b}], {n, 20}]
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CROSSREFS
| Cf. A000537.
Sequence in context: A050181 A089931 A148946 * A148947 A090573 A048119
Adjacent sequences: A096219 A096220 A096221 * A096223 A096224 A096225
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KEYWORD
| nonn
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AUTHOR
| Bill Liebeskind (billlieb(AT)hotmail.com), Jul 29 2004
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EXTENSIONS
| Edited by T. D. Noe (noe(AT)sspectra.com), Jul 30 2004
More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 31 2004
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