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Factoriangular number

In number theory, a factoriangular number is an integer formed by adding a factorial and a triangular number with the same index. The name is a portmanteau of "factorial" and "triangular."

Definition

For , the th factoriangular number, denoted , is defined as the sum of the th factorial and the th triangular number:

.

The first few factoriangular numbers are:

These numbers form the integer sequence A101292 in the Online Encyclopedia of Integer Sequences (OEIS).

Properties

Recurrence relations

Factoriangular numbers satisfy several recurrence relations. For ,

And for ,

These are linear non-homogeneous recurrence relations with variable coefficients of order 1.

Generating functions

The exponential generating function for factoriangular numbers is (for )

If the sequence is extended to include , then the exponential generating function becomes

.

Representations as sums of triangular numbers

Factoriangular numbers can sometimes be expressed as sums of two triangular numbers:

  • if and only if or .
  • if and only if is a perfect square. For , the only known solution is , giving .
  • if and only if is a sum of two squares.

Representations as sums of squares

Some factoriangular numbers can be expressed as the sum of two squares. For , the factoriangular numbers that can be written as for some integers and include:

This result is related to the sum of two squares theorem, which states that a positive integer can be expressed as a sum of two squares if and only if its prime factorization contains no prime factor of the form raised to an odd power.

Fibonacci factoriangular numbers

A Fibonacci factoriangular number is a number that is both a Fibonacci number and a factoriangular number. There are exactly three such numbers:

This result was conjectured by Romer Castillo and later proved by Ruiz and Luca.

Pell factoriangular numbers

A Pell factoriangular number is a number that is both a Pell number and a factoriangular number. Luca and Gómez-Ruiz proved that there are exactly three such numbers: , , and .

Catalan factoriangular numbers

A Catalan factoriangular number is a number that is both a Catalan number and a factoriangular number.

Generalizations

The concept of factoriangular numbers can be generalized to -factoriangular numbers, defined as where and are positive integers. The original factoriangular numbers correspond to the case where . This generalization gives rise to factoriangular triangles, which are Pascal-like triangular arrays of numbers. Two such triangles can be formed:

  • A triangle with entries where , yielding the sequence: 2, 3, 5, 7, 9, 12, 25, 27, 30, 34, ...
  • A triangle with entries where , yielding the sequence: 2, 4, 5, 7, 8, 12, 11, 12, 16, 34, ...

In both cases, the diagonal entries (where ) correspond to the original factoriangular numbers.

See also

References

External links