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.

Definition

The boustrophedon transform is a numerical, sequence-generating transformation, which is determined by a binary operation such as addition.

. The entirety of the transformation itself can be visualized (or imagined) as being constructed by filling-out the triangle as shown in Figure 1.

Boustrophedon Triangle

To fill-out the numerical Isosceles triangle (Figure 1, and place one value (from the input sequence) per row, using the boustrophedon scan (zigzag- or serpentine-like) approach.

, and we number this top row as row 0.

) as follows:

• values in the row.

on the right-hand end of the row.

• ).

• is odd).

on the left-hand end of the row.

• ).

• is even).

Refer to the arrows in Figure 1 for a visual representation of these "addition" operations.

.

Recurrence relation

A more formal definition uses a recurrence relation (with kn0) by

1. .

and greater indices).

pairs:

Special Cases

In the case a0 = 1, an = 0 (n 0), the resulting triangle is called the SeidelEntringerArnold Triangle 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.

Algebraic definition(s)

) can be defined for different algebras ("numeric domains").

Euclidean (Real) values

)-valued scalars, the boustrophedon transformed Real-value is related to the input value, , as:

,

with the reverse relationship (input from output) defined as:

,

where is the sequence of "up/down" numbers—also known as secant or tangent numbers.

The exponential generating function

The exponential generating function of a sequence (an) is defined by

The exponential generating function of the boustrophedon transform (bn) is related to that of the original sequence (an) by

The exponential generating function of the unit sequence is1, so that of the up/down numbers is secx+tanx.

References