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# Boustrophedon transform

The Boustrophedon transform is a sequence transform which preserves integer sequences. The B-transform of an integer sequence is most simply defined as , introducing an auxiliary triangle defined by

## Contents

## Practical calculation

The above defining formula can be applied as follows: To calculate the triangle , fill the increasingly longer rows in alternating directions (one row from left to right, next row from right to left), starting the n-th row at the respective beginning with , and then adding as next element the sum of the previously placed element plus the nearest element on the previous row:

T00 := a0 a1 =: T10 --------> T11=T10+T00 T22=T21+T10 <-------- T21=T20+T11 <-------- T20 := a2 a3 =: T30 --------> T31=T30+T22 --------> T32=T31+T21 --------> T33=T32+T20 (...) <-------- T41=T40+T33 <-------- T40 := a4

and so on.

The "final" element of each row is defined to be , so

### Explicit formula

As a result of the construction, one has , where the **Boustrophedon coefficients** A109449 are given by

with the usual binomial coefficients and the so-called *Euler numbers* given in A000111, with e.g.f. .

The table of these Boustrophedon coefficients reads:

1; 1, 1; 1, 2, 1; 2, 3, 3, 1; 5, 8, 6, 4, 1; 16, 25, 20, 10, 5, 1; 61, 96, 75, 40, 15, 6, 1; (...)

If this table is considered as an infinite square array or matrix **B**, completing the upper right half with zeros, the Boustrophedon transform can be expressed as simply multiplying the sequence, considered as a column matrix, by this matrix **B**.

### Inverse transform

The above infinite matrix **B** has an inverse (A247453) which is equal to **B** up to alternating signs :

As a consequence, multiplication by this inverse matrix is the inverse transformation, which can also be expressed as

## History

*(to be written)* -- see the Millar, Sloane & Young reference.

## Programs

### PARI/gp

B(a,T=a[1..1],S)={for(n=2,#a,T=vector(n,k,if(k>1,S+=T[n-k+1],S=a[n]))/*;print(T)*/;a[n]=S);a}

or

B(a)={a+vector(#a,n,sum(k=1,n-1,abs(polylog(k-n,I))*2*binomial(n-1,k-1)*a[k]))}

See the "Transformations of Integer Sequences" page on main OEIS web site for further Maple, Mathematica and PARI/GP code.

## References

- J. Millar, N. J. A. Sloane and N. E. Young,
*A new operation on sequences: the Boustrophedon transform,*J. Comb. Theory**17A**(1996) pp. 44-54.

## Authorship

This page was created by M. F. Hasler on Oct. 5, 2017.

### Cite this page as

M. F. Hasler, *Boustrophedon transform*.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Boustrophedon_transform]