Бұрысты жолмен трансформациялау немесе тізбектерді өңдеу әдісі

Mathematical transformation on sequences
In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag or serpentine like) manner—as opposed to a "Raster Scan" sawtooth like manner.

The boustrophedon transform is a numerical, sequence generating transformation, which is determined by a binary operation such as addition. Generally speaking, given a sequence: , the boustrophedon transform yields another sequence: , where is likely defined equivalent to The entirety of the transformation itself can be visualized (or imagined) as being constructed by filling out the triangle as shown in Figure 1.

To fill out the numerical Isosceles triangle (Figure 1), you start with the input sequence, , and place one value (from the input sequence) per row, using the boustrophedon scan (zigzag or serpentine like) approach. The top vertex of the triangle will be the input value , equivalent to output value , and we number this top row as row 0. The subsequent rows (going down to the base of the triangle) are numbered consecutively (from 0) as integers—let denote the number of the row currently being filled. These rows are constructed according to the row number as follows:
For all rows, numbered , there will be exactly values in the row. If is odd, then put the value on the right hand end of the row. Fill out the interior of this row from right to left, where each value (index: ) is the result of "addition" between the value to right (index: ) and the value to the upper right (index: ). The output value will be on the left hand end of an odd row (where is odd). If is even, then put the input value on the left hand end of the row. Fill out the interior of this row from left to right, where each value (index: ) is the result of "addition" between the value to its left (index: ) and the value to its upper left (index: ). The output value will be on the right hand end of an even row (where is even). Refer to the arrows in Figure 1 for a visual representation of these "addition" operations. For a given, finite input sequence: , of values, there will be exactly rows in the triangle, such that is an integer in the range: (exclusive). In other words, the last row is .

In the case a0 = 1, an = 0 (n > 0), the resulting triangle is called the Seidel–Entringer–Arnold Triangle and the numbers are called Entringer numbers
In this case the numbers in the transformed sequence bn are called the Euler up/down numbers. This is sequence A000111 on the On Line Encyclopedia of Integer Sequences. These enumerate the number of alternating permutations on n letters and are related to the Euler numbers and the Bernoulli numbers.

Building from the geometric design of the boustrophedon transform, algebraic definitions of the relationship from input values to output values can be defined for different algebras ("numeric domains").