Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4 |

Department of Mathematics

University of Evansville

1800 Lincoln Avenue

Evansville, IN 47722

USA

**Abstract:**

Increasing sequences *a*( ) and
*b*( ) that partition the sequence of positive
integers are called complementary sequences, and equations that explicitly
involve both *a*( ) and *b*( ) are called complementary equations.
This article
surveys several families of such equations,
including *b*(*n*) = *a*(*jn*) +- *r*,
*b*(*n*) = *a*(*jn*)+*kn*,
*b*(*n*) = *f*(*a*(*n*)),
and *b*(*n*) = *a*(*b*(*n*-1))+*qn*+*r*.

(Concerned with sequences A000124 A000201 A000217 A000290 A000326 A000384 A001844 A001950 A002061 A003159 A005228 A028387 A036554 A045671 A045672 A045681 A045749 A045750 A045774 A045775 A058331 and A080164 .)

Received May 16 2006;
revised versions received July 26 2006; October 11 2006.
Published in *Journal of Integer Sequences* December 30 2006.

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