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Leonardo of Pisa (c. 1170 – c. 1250) described this method [1] [2] for generating primitive triples using the sequence of consecutive odd integers ,,,,, … and the fact that the sum of the first n terms of this sequence is .
Exactly one of a, b, c is divisible by 5. [8] The largest number that always divides abc is 60. [15] Any odd number of the form 2m+1, where m is an integer and m>1, can be the odd leg of a primitive Pythagorean triple. See almost-isosceles primitive Pythagorean triples section below. However, only even numbers divisible by 4 can be the even leg ...
The sum of the reciprocals of the palindromic numbers converges to approximately 3.3703 . A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the five-term row 1 4 6 4 1 . The sum of the reciprocals of the pentatope numbers is 4 / 3 .
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
The number of binary strings of length n without an odd number of consecutive 1 s is the Fibonacci number F n+1. For example, out of the 16 binary strings of length 4, there are F 5 = 5 without an odd number of consecutive 1 s—they are 0000, 0011, 0110, 1100, 1111.
Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. Well, one of those three possibilities for odd numbers causes an issue.
The numerator equates to the number of ways to select the winning numbers multiplied by the number of ways to select the losing numbers. For a score of n (for example, if 3 choices match three of the 6 balls drawn, then n = 3), ( 6 n ) {\displaystyle {6 \choose n}} describes the odds of selecting n winning numbers from the 6 winning numbers.
The function q(n) gives the number of these strict partitions of the given sum n. For example, q(3) = 2 because the partitions 3 and 1 + 2 are strict, while the third partition 1 + 1 + 1 of 3 has repeated parts. The number q(n) is also equal to the number of partitions of n in which only odd summands are permitted. [20]