Doubly triangular number

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Summing up to the n-th row of Floyd's triangle yields the n-th doubly triangular number Floyd triangle doubly triangular number.svg
Summing up to the n-th row of Floyd's triangle yields the n-th doubly triangular number
There are 21 colorings of the four corners of a square using three colors (up to symmetry), a doubly triangular number, formed by combining two of the six colorings of two opposite corners Square 3-colorings.svg
There are 21 colorings of the four corners of a square using three colors (up to symmetry), a doubly triangular number, formed by combining two of the six colorings of two opposite corners

In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if denotes the th triangular number, then the doubly triangular numbers are the numbers of the form .

Contents

Sequence and formula

The doubly triangular numbers form the sequence [1]

0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, ...

The th doubly triangular number is given by the quartic formula [2]

The sums of row sums of Floyd's triangle give the doubly triangular numbers. Another way of expressing this fact is that the sum of all of the numbers in the first rows of Floyd's triangle is the th doubly triangular number. [1] [2]

Sum of reciprocals

A formula for the sum of the reciprocals of the doubly triangular numbers is given by

In combinatorial enumeration

Doubly triangular numbers arise naturally as numbers of unordered pairs of unordered pairs of objects, including pairs where both objects are the same:

When pairs with both objects the same are excluded, a different sequence arises, the tritriangular numbers which are given by the formula . [5]

In numerology

Some numerologists and biblical studies scholars consider it significant that 666, the number of the beast, is a doubly triangular number. [6] [7]

References

  1. 1 2 3 Sloane, N. J. A. (ed.), "SequenceA002817(Doubly triangular numbers)", The On-Line Encyclopedia of Integer Sequences , OEIS Foundation
  2. 1 2 Gulliver, T. Aaron (2002), "Sequences from squares of integers", International Mathematical Journal, 1 (4): 323–332, MR   1846748
  3. Barnett, Michael P. (2003), "Molecular integrals and information processing", International Journal of Quantum Chemistry, 95 (6), Wiley: 791–805, doi:10.1002/qua.10614
  4. Mathar, Richard J. (2017), Statistics on small graphs, row 2 of table 60, arXiv: 1709.09000
  5. Sloane, N. J. A. (ed.), "SequenceA050534(Tritriangular numbers)", The On-Line Encyclopedia of Integer Sequences , OEIS Foundation
  6. Watt, W. C. (1989), "666", Semiotica, 77 (4), doi:10.1515/semi.1989.77.4.369, S2CID   263854723
  7. Heick, Otto William (January 1985), "The Antichrist in the Book of Revelation", Consensus, 11 (1), Article 3