For two integers a and b, does anyone know what the formula a[sup]2[/sup] - a = b, for b = 0, 2, 6, 12, 20, 30, 42, is called? Or if the sequence has a name? ... if instead of making b a function of the integer value of a, you make [the domain] a function of b and start with b = 1, the only positive solution is the golden ratio, so the sequence is a rational function of phi, but is it called that?
No cigar. It matches except for the zero, what to do? I guess a gold star will have to do. Ah, I see. If I make a equal 1 by substituting Phi[sup]2[/sup] - Phi for 1, I'm doing what, projecting Phi onto the integers? Then I have [ Phi[sup]2[/sup] - Phi ] [sup]2[/sup] - Phi[sup]2[/sup] - Phi = 0, right?
I'm on it. If you take the 7th number in the sequence and show it has the form a[sup]2[/sup] - a, then you have a span of numbers between b = 30 and b = 42, in the golden "field" isn't it? Which is taking the 6th triangular number, and the Catalan numbers along?
Because when the "number", which I will call a packing, is 30, it's the square of a product of the first two primes minus the product. Out falls 2x3x5, a product with a factor which is the third prime. which is if you will, a kind of "number triangle"
James, I'm so glad we get to have these little chats. What I think I'm trying to do is understand a sequence, that doesn't look symmetrical, in terms of other sequences which have a symmetry. This sequence starts with the first four triangular numbers then it runs into a "packing constraint". The 3rd triangular number is 6, which is 3[sup]2[/sup] - 3 = ( 3[sup]2[/sup] + 3 )/2; 6[sup]2[/sup] - 6 != ( 6[sup]2[/sup] + 6 )/2. This sequence I have next uses T[sub]5[/sub] - 1, T[sub]6[/sub] - 2, and T[sub]7[/sub] - 3. I'm just trying to understand it.
http://en.wikipedia.org/wiki/Pronic_number The first reply in this thread was also correct, if you start from n= 0,1,2... , this should have been obvious since, if we call the Pronic numbers \( P_1,P_2, \ldots \) and the triangular numbers \( T_i \) , we have \( T_n = \frac{n (n+1)}{2} \Rightarrow T_{n-1} = \frac{(n-1)^2 + n - 1}{2} = \frac{n^2-n}{2} = \frac{1}{2}P_n \) \( \Rightarrow P_n = 2 T_{n-1} \) I have no idea what the hell you are talking about in the rest of this thread.
I guess what the hell I'm talking about is this: It has always been possible to rotate a 3-dimensional object--at least, since there has been 3 dimensions--if there is a 4th dimension to "do it" in. If you imagine that since the "dawn" of 4-dimensional spacetime there has always been somewhere in it that a stack of 8 "cubes" of space can rotate with respect to each other, then it must be that there is a way to rotate them in 2 of the 3, or 1 of 2, or none, simultaneously. The difference is something a Rubik's cube tells you about--a rotation in the x, followed by one in the y dimension in the same direction, is different to a xy rotation simultaneously. Continous transformations have the same result, but discontinuous ones don't, since xy and yx "one dimension at a time" are different, but "two at a time" they look the same. Because of the shortest path problem, we know the dimensions of the 2x2x2 "rotation space" and that it has a limit to how many "free" rotations you can make (without choosing a direction), before having to choose the direction--left or right handed rotations are equivalent if you are "free" to choose; after so many of these the free ride is over. Rotations have "time" attached to them, so all the positions in the space are fixed points and independent of time. Surprisingly, or maybe not so much, if you do choose a direction first (call it t+, if you want) you only have 6 free (undirected) rotations before having to choose "the direction of" t again.
1) Integer Sequence 2) Something about the golden ratio 3) Some nonsense about projecting onto the integers 4) Golden field :shrug: 5) Something about prime factors 6) ????? 7) RUBIK'S CUBE!! This is like watching an episode of Family Guy, none of the jokes seem to follow the main story line and in the end it just seems a little try hard and pathetic.
It's a good thing then, that this forum has people like you who are rational enough to not make comments that look pathetic, or like they just had to post something because they got bored. Not you though, if you don't see the point of someone's post you make intelligent and encouraging noises, about things you've never heard of. It must be really nice to be so intelligent.
The golden ratio \( \Phi \;= \;\frac {1 + \sqrt 5} {2} \), can be considered as a number like \( x + y\sqrt 5, \;x = y = \frac 1 2 \). If \( x\; \ne \;y \) you have a number in the "golden field". So the golden ratio is one of these numbers, and any number in the field with x = y is a factor of \( \Phi \).
A number can be considered a number? Whatever next! With regard these threads, I've said it before and I'll say it again: lock and ban.
Well, a number can be considered as a number which is like another number. Like the number 1 is like 1/2 of the number 2. What are these mathematicians going to dream up next? I've said it before and I'll say it again, Guest254 needs to find something to do, like get a life.
