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Expanding (x + y) n yields the sum of the 2 n products of the form e 1 e 2... e n where each e i is x or y. Rearranging factors shows that each product equals x n−k y k for some k between 0 and n. For a given k, the following are proved equal in succession: the number of terms equal to x n−k y k in the expansion
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.
Express each term of the final sequence y 0, y 1, y 2, ... as the sum of up to two terms of these intermediate sequences: y 0 = x 0, y 1 = z 0, y 2 = z 0 + x 2, y 3 = w 1, etc. After the first value, each successive number y i is either copied from a position half as far through the w sequence, or is the previous value added to one value in the ...
On the other hand, if such a tiling uses exactly k of the 2 × 1 tiles, then it uses n − 2k of the 1 × 1 tiles, and so uses n − k tiles total. There are ( n − k k ) {\displaystyle {\tbinom {n-k}{k}}} ways to order these tiles, and so summing this coefficient over all possible values of k gives the identity.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
2 + (1 + 3) = (2 + 1) + 3 with segmented rods. Addition is associative, which means that when three or more numbers are added together, the order of operations does not change the result. As an example, should the expression a + b + c be defined to mean (a + b) + c or a + (b + c)? Given that addition is associative, the choice of definition is ...
If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares. (This is Euler's first Proposition). Indeed, suppose for example that a 2 + b 2 {\displaystyle a^{2}+b^{2}} is divisible by p 2 + q 2 {\displaystyle p^{2}+q^{2}} and that this latter is a prime.
Pursuing this type of analysis more carefully, G. H. Hardy and John Edensor Littlewood in 1923 conjectured (as part of their Hardy–Littlewood prime tuple conjecture) that for any fixed c ≥ 2, the number of representations of a large integer n as the sum of c primes n = p 1 + ⋯ + p c with p 1 ≤ ⋯ ≤ p c should be asymptotically equal to