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from the formula for the tangent of the difference of angles. Using s instead of r in the above formulas will give the same primitive Pythagorean triple but with a and b swapped. Note that r and s can be reconstructed from a, b, and c using r = a / (b + c) and s = b / (a + c).
Then = {Thus, for instance, = =: dividing 12 by two, twice, leaves the odd number 3. As another example, t 13 = 1 {\displaystyle t_{13}=1} because 13 is congruent to 1 mod 4. The paperfolding word 1101100111001001..., which is created by concatenating the terms of the regular paperfolding sequence, is a fixed point of the morphism or string ...
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 ...
This follows, for instance, from Maekawa's theorem, which states that the number of mountain folds at a flat-folded vertex differs from the number of valley folds by exactly two folds. [3] Therefore, suppose that a crease pattern consists of an even number 2 n of creases, and let α 1 , α 2 , ⋯, α 2 n be the consecutive angles between the ...
For example, the first four odd numbers form the sequence (1, 3, 5, 7). ... If each consecutive ... is a sequence of complex numbers rather than a sequence of real ...
If a representation has an odd number of terms, x/y is the middle term, while if it has an even number of terms and its minimum value is m it may be extended in a unique way to a longer sequence with the same sum and an odd number of terms, by including the 2m − 1 numbers −(m − 1), −(m − 2), ..., −1, 0, 1, ..., m − 2, m − 1.
A sequence of (u, v)-Ulam numbers is regular if the sequence of differences between consecutive numbers in the sequence is eventually periodic. When v is an odd number greater than three, the (2, v)-Ulam numbers are regular. When v is congruent to 1 (mod 4) and at least five, the (4, v)-Ulam numbers are again regular. However, the Ulam numbers ...
For example, the diagram below shows n = 20 and the partition 20 = 7 + 6 + 4 + 3. Let m be the number of elements in the smallest row of the diagram (m = 3 in the above example). Let s be the number of elements in the rightmost 45 degree line of the diagram (s = 2 dots in red above, since 7