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A level-5 approximation to a Sierpiński triangle obtained by shading the first 2 5 (32) levels of a Pascal's triangle white if the binomial coefficient is even and black otherwise If one takes Pascal's triangle with 2 n {\displaystyle 2^{n}} rows and colors the even numbers white, and the odd numbers black, the result is an approximation to ...
In their magic triangles, the sum of the k-th row and the (n-k+1)-th row is same for all k. [5] (sequence A356808 in the OEIS) Its one modification uses triangular numbers instead of square numbers. (sequence A355119 in the OEIS) Another magic triangle form is magic triangles with triangular numbers with different summation. In this magic ...
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a ...
A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). [1] For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not.
Visual proof for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BCE Oracle bone script numeral system counting rod place value decimal. Shang dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the I Ching, which greatly influenced written literature during the Zhou dynasty (1050–256 BC).
In number theory, a pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, either from left to right or from right to left. It is named because it represents the number of 3-dimensional unit spheres which can be packed into a pentatope (a 4-dimensional tetrahedron ) of increasing ...
It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos −1 ( 1 / 5 ), or approximately 78.46°. The 5-simplex is a solution to the problem: Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010).The n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.
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