Boustrophedon transform

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. 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 an 'addition' operation. Generally speaking, given a sequence: ( a 0 , a 1 , a 2 , … ) {displaystyle (a_{0},a_{1},a_{2},ldots )} , the boustrophedon transform yields another sequence: ( b 0 , b 1 , b 2 , … ) {displaystyle (b_{0},b_{1},b_{2},ldots )} , where b 0 {displaystyle b_{0}} is likely defined equivalent to a 0 {displaystyle a_{0}} . 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, ( a 0 , a 1 , a 2 , … ) {displaystyle (a_{0},a_{1},a_{2},ldots )} , 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 a 0 {displaystyle a_{0}} , equivalent to output value b 0 {displaystyle b_{0}} , 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 k {displaystyle k} denote the number of the row currently being filled. These rows are constructed according to the row number ( k {displaystyle k} ) as follows: Refer to the arrows in Figure 1 for a visual representation of these 'addition' operations. Note that for a given, finite input-sequence: ( a 0 , a 1 , . . . a N ) {displaystyle (a_{0},a_{1},...a_{N})} , of N {displaystyle N} values, there will be exactly N {displaystyle N} rows in the triangle, such that k {displaystyle k} is an integer in the range: [ 0 , N ) {displaystyle [0,N)} (exclusive). In other words, the last row is k = N − 1 {displaystyle k=N-1} . A more formal definition uses a recurrence relation. Define the numbers T k , n {displaystyle T_{k,n}} (with k ≥ n ≥ 0) by