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Search: id:A133257
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| A133257 |
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The seven terms in the preceding field are, in symbolic notation, [a(0), a(1), a(2), a(3), a(4), a(5), a(6),] in an infinite sequence. Each integer in this sequence a(n-1) is the sum of the total number of edges of a rectangular sheet folded "n-1" times, starting at n=1 for zero folds. I.e., n=1 for the originally unfolded sheet. Each fold halves the area of the sheet, producing two more rectangles of equal area. The angle each (straight) fold line makes with the long dimension of the original rectangle alternates in degrees as 90, 0, 90, 0, etc. with fold order, with the angle for the first fold (n-1=1) having the non-unique convention of being 90 degrees. |
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1
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OFFSET
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0,1
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COMMENT
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Each number [a(0), a(1), a(2), a(3), ...] in the sequence was generated by physical experiment, i.e., by actually folding the sheet in the manner described, while counting the number of edges for each of the four sides of the thus folded sheet, and then summing the total number of edges for a given fold order. An edge can be an edge of the original sheet, or can be a folded edge. For convenience, the edge count for all sides for an [n-1] order fold proceeded via rotation (for concreteness) about a vertical axis of the folded sheet by 90 degree increments, with the convention (when assembling a multi-column table to keep track of counts and count sums) of starting with the side of a folded rectangle with the single edge (always a folded edge for n>0), and with the next edge count always being taken as the one which equals two (regardless of order of fold), and then with the other two sides. The line of sight for making each visual observation was perpendicular to a given edge, while being parallel to the 'plane' of the folded rectangle. The area of this rectangle, A(n), is given by A(n)= A(0)/2^(n-1). The diminishing area and bulk of the (master) fold with monotonically increasing order of fold limited the value of n at which a reliable measurement (count) could be made from visual observation for a standard document paper size (8 1/2'' by 11").
All necessary instructions are provided in this submission for generating and confirming the sequence provided, which has been checked and rechecked for its first seven terms.
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FORMULA
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See "Comments" for the operational definition for determining the value of a given integer in this sequence. Also, the algebraic formula for the general case is derivable via use of the rules of this well-defined operation, plus mathematical induction to prove that the algorithm for the sequence, if true for some given "n" is then true for the [n+1]th term. It is unknown if the formula is that of a recurrence relation, whether linear or nonlinear.
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EXAMPLE
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a(0)=4, a(4)=25.
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CROSSREFS
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Cf. A014577, A014707.
Adjacent sequences: A133254 A133255 A133256 this_sequence A133258 A133259 A133260
Sequence in context: A097403 A126618 A049648 this_sequence A023666 A023502 A024882
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KEYWORD
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nonn,uned
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AUTHOR
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Harold M. Frost, III (halfrost(AT)charter.net), Dec 19 2007
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