Tetrahedral numberA tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron.
Integer overflowIn computer programming, an integer overflow occurs when an arithmetic operation on integers attempts to create a numeric value that is outside of the range that can be represented in the space allocated for the result – either higher than the maximum or lower than the minimum representable value. Most integer arithmetic in modern computation uses binary representation of integers, though decimal representation also exists.
Binomial coefficientIn mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written (nk){\displaystyle {\tbinom {n}{k}}} or C(n,k){\displaystyle C(n,k)}.
Pentagonal numberA pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex.
Mathematical inductionMathematical induction is a method for proving that a statement P ( n ) {\displaystyle P(n)} is true for every natural number n {\displaystyle n} , that is, that the infinitely many cases P ( 0 ) , P ( 1 ) , P ( 2 ) , P ( 3 ) , … {\displaystyle P(0),P(1),P(2),P(3),\dots } all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true.
Equilateral triangleShape with three equal sidesAn equilateral triangle is a triangle with three sides of equal length and three equal angles. It is a regular polygon, occasionally known as the regular triangle. It is a special case of an isosceles triangle by modern definition. Its internal angle is 60° in degree or π/3 in radian. Its perimeter is three times the side length. Its area is a quarter of the square root of three multiplied with its squared side length.
On-Line Encyclopedia of Integer SequencesThe On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited.
Figurate numberThe term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers).
DicuilDicuilus (Gaelic: Dícuil; fl.814–825 A.D.) was an Irish monk, astronomer, geographer and author born during the second half of the 8th century, possibly in the Hebrides. He travelled the Frankia around the turn of the 9th century and was involved with the Carolingian Renaissance under Louis the Pious. He was the author of astronomical and cosmographical treatises during the early 9th century, an example of Hiberno-Latin culture.
Carl Friedrich GaussGerman polymath and scholar (1777–1855)Johann Carl Friedrich Gauss ( ; German: Gauß; 30 April 1777 – 23 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. His mathematical contributions spanned the branches of number theory, algebra, analysis, geometry, statistics, and probability.
Proof without wordsIn mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.Empty sumIn mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the additive identity. Let a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , a 3 {\displaystyle a_{3}} , ... be a sequence of numbers, and let s m = ∑ i = 1 m a i = a 1 + ⋯ + a m {\displaystyle s_{m}=\sum _{i=1}^{m}a_{i}=a_{1}+\cdots +a_{m}} be the sum of the first m terms of the sequence.*As an Amazon Associate I earn from qualifying purchases.