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triangular numbers Triangular numbers are defined as and are among the simplest figurate numbers (see picture aside).

Gauss proved that every number is the sum of at most 3 triangular numbers. Ming showed that the only triangular numbers which are also Fibonacci numbers are 1, 3, 21 and 55.

N.Tzanahis B.M.M. de Weger proved that there are six which are equal to the product of 3 consecutive integers: , , , , and The first nontrivial palindromic with palindromic index are , , and . The largest know such number, found by P.De Geest, is .

There are several interesting formulas involving triangular numbers, for example: The sum of the reciprocals of triangular number is 2.

By solving the diophantine equation , it is easy to find infinite triangular numbers which are also square. The first are 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056,...

The 3 smallest primes which concatenated with the previous prime gives a triangular number are 1812341, 624403308264975451 and 48127684695939820823. For this 3rd value we have Below, the spiral pattern of triangular numbers up to . See the page on prime numbers for an explanation and links to similar pictures. The first triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300 more terms

Triangular numbers can also be... (you may click on names or numbers and on + to get more values)