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A Pell–Wieferich prime is a prime p satisfying p 2 divides P p−1, when p congruent to 1 or 7 (mod 8), or p 2 divides P p+1, when p congruent to 3 or 5 (mod 8), where P n denotes the n-th Pell number. For example, 13, 31, and 1546463 are Pell–Wieferich primes, and no others below 10 9 (sequence A238736 in the OEIS). In fact, Pell ...
The number of different beam paths that have k reflections, for k > 1, is the k-th Fibonacci number. (However, when k = 1, there are three reflection paths, not two, one for each of the three surfaces.) [93] Fibonacci retracement levels are widely used in technical analysis for financial market trading.
A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47.
"subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.
The th fibbinary number (counting 0 as the 0th number) can be calculated by expressing in its Zeckendorf representation, and re-interpreting the resulting binary sequence as the binary representation of a number. [1] For instance, the Zeckendorf representation of 19 is 101001 (where the 1's mark the positions of the Fibonacci numbers used in ...
If the number subtracted was the i th Fibonacci number F(i), put a 1 in place i − 2 in the code word (counting the left most digit as place 0). Repeat the previous steps, substituting the remainder for N, until a remainder of 0 is reached. Place an additional 1 after the rightmost digit in the code word.
A digit sequence with rank r may be formed either by adding the digit 2 to a sequence with rank r − 2, or by adding the digit 1 to a sequence with rank r − 1.If f is the function that maps r to the number of different digit sequences of that rank, therefore, f satisfies the recurrence relation f (r) = f (r − 2) + f (r − 1) defining the Fibonacci numbers, but with slightly different ...
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