Patterns,from,the,Diagonals,Pa education Patterns from the Diagonals of Pascals Triangle
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Common sequences whichare discussed in Pascal's Triangle include the counting numbers and trianglenumbers from the diagonals of Pascal's Triangle. By examining these diagonals,however, not only do we find these two sequences, but a whole shower ofsequences, which appear to get ever more complicated, each one a development ofthe last one. In this article, I discuss these sequences and how they arelinked to each other.Sequences can be found in the diagonals of Pascal's triangle. In the nextdiagram, however, as it is left justified, we need to carefully examine thecolumns, and see what sequences we can spot.1 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 1Working our way inwards, in the first column, the numbers are always 1.Next, we have the counting numbers in order in the second column. Then, we havethe slightly more complicated sequence of the Triangle numbers (These are sumsof all previous counting numbers - for example, the 8th triangle number is1+2+3+4+5+6+7+8 = 36. Also, by creating triangles of dots or counters, andcounting the number of these dots used, you can get the triangle numbers) Ifyou take the differences of consecutive triangle numbers, you get the countingnumbers. This is shown below:Sequence: 1 3 6 10 15 21 28Differences: 2 3 4 5 6 7The sequence in the fourth column is more complicated again. The numbers1,4,10,20,35 are called tetrahedral numbers. Like triangle numbers, these canbe understood visually. This time, you need to create triangular based pyramids(NOT square based pyramids like those in Egypt). For your pyramids, you coulduse coins, counters, marbles - whatever you like. Like before, all you have todo is count the number of things used to create your pyramid. Also, wecan also express the tetrahedral numbers in a "difference tree".1 4 10 20 35 563 6 10 15 21We can begin to see an interesting pattern emerging. The differences of thecounting numbers 1,2,3,4,5,6,7,8... from column two of Pascal's triangle arealways one, and the first column of Pascal's triangle is also always 1s.Similarly, the triangle numbers 1,3,6,10,15,21... are from column 3 and givedifferences of 1,2,3,4,5,6... , which is column two, and the tetrahedralnumbers from the fourth row (1,4,10,20,35... ) have differences of the trianglenumbers from the third row of the triangle.In fact, this pattern always continues. The differences of one column givesthe numbers from the previous column (the first number 1 is knocked off,however). So, for example, if we look at the fifth column in Pascal's triangle,we get the sequence 1,5,15,35,70,126... These are called the pentatope numbers,and appear to be a very complicated sequence. However, their differences justgive the tetrahedral numbers, (starting from 4).It is clear, therefore, that Pascal's triangle is a powerful tool in makingsense of these complicated sequences, and contains patterns in its diagonalswhich are far more extensive than one might intitially imagine. Normal 0 false false false EN-GB X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-qformat:yes;mso-style-parent:"";mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-para-margin-top:0cm;mso-para-margin-right:0cm;mso-para-margin-bottom:10.0pt;mso-para-margin-left:0cm;line-height:115%;mso-pagination:widow-orphan;font-size:11.0pt;font-family:"Calibri","sans-serif";mso-ascii-font-family:Calibri;mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:minor-fareast;mso-hansi-font-family:Calibri;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi;}
Patterns,from,the,Diagonals,Pa