Pentagonalrekurs

Pentagonalrekurs, or pentagonal recursion, is a term used in some mathematical discussions to denote a family of recursive schemes whose increments are governed by pentagonal-number patterns. In its simplest instantiation, it defines a sequence by a(0)=0 and a(n)=a(n-1) + 3n - 2 for n≥1. This yields the pentagonal numbers P(n) = n(3n-1)/2, with initial values 0, 1, 5, 12, 22, 35, etc.

Variants of pentagonalrekurs appear in related areas such as partition theory and polygonal number sequences. A common generalization replaces the fixed increment 3n-2 with an alternative function f(n) that encodes pentagonal structure, for example allowing sign changes or incorporating generalized pentagonal numbers P(k) = k(3k-1)/2 for k in Z. Such recurrences generate sequences with quadratic growth and can be used as lightweight generators of pentagonal-like sequences in algorithmic contexts.

The concept is connected to broader ideas in number theory and combinatorics, including Euler’s pentagonal number theorem and the study of polygonal numbers. As a teaching and research tool, pentagonalrekurs offers a simple framework for exploring how recurrence relations can reproduce classical numerical patterns and support efficient term generation for larger sequences.

See also: pentagonal numbers, Euler’s pentagonal number theorem, polygonal numbers, recurrence relations.