These are the successive triangular numbers
formed from the sequence:
The three lessons are based on investigations into triangular numbers
, providing an alternative to the linear functions work we generally see in this area.
The author covers Pythagorean triples, the Monty Hall problem and other probability theory puzzles, the Lucas sequence, the triangular numbers
, the square numbers, and many other related subjects.
We also give an application of the proposed fuzzy ordering method to the fuzzy risk analysis problem, where the evaluating values are represented by triangular numbers
In this method according to the number of options the membership function is considered as the following illustration and triangular numbers
are also determined in this way.
The squares make up the perfect pangram I'D PAY, but despite a good consonant-vowel balance it is difficult to get anything out of the triangular numbers
, for which the relevant letters are ACFJOU.
Similarly to right and left triangular numbers
we can introduce right and left trapezoidal numbers as parts of a trapezoidal number.
A mate went for the same effect, but his odd triangular numbers
looked like bushy wedges of Dairylea.
Well done to Dominic Degnan from County Durham who correctly worked out that 1, 3, 6, 10, 15, 21 are triangular numbers
in last fortnight's quiz.
The second vector is the sum total of triangular numbers
of the above matrix as inversed.
The triangular numbers
are formed by partial sum of the series 1 + 2 + 3 + 4 + 5 + 6 + 7.
Torres and Viorica Teca  have further investigated these sequences and defined mirror and symmetric Smarandache sequences of Triangular numbers
making use of Maple system.