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In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted F n .
For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. p n is given by the formula: = = + for n ≥ 1. The first few pentagonal numbers are:
The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9". The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.
As with the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediately previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are L 0 = 2 {\displaystyle L_{0}=2} and L 1 = 1 {\displaystyle L_{1}=1} , which differs from the first two Fibonacci numbers F 0 = 0 {\displaystyle F_{0}=0 ...
The product of the members of a finite arithmetic progression with an initial element a 1, common differences d, and n elements in total is determined in a closed expression a 1 a 2 a 3 ⋯ a n = a 1 ( a 1 + d ) ( a 1 + 2 d ) . . .
Diagram illustrating three basic geometric sequences of the pattern 1(r n−1) up to 6 iterations deep.The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.
Then the triangle is in Euclidean space if the sum of the reciprocals of p, q, and r equals 1, spherical space if that sum is greater than 1, and hyperbolic space if the sum is less than 1. A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270.
Layers of Pascal's pyramid derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial – the number of terms is clearly a triangular number. In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by