Is there a pattern to 0,6,24,60,120,210? This is fiendishly hard. Or how about 0,4,10,20,35. Please save me and solve!
+26393 Is there a pattern to 0,6,24,60,120,210? This is fiendishly hard. Or how about 0,4,10,20,35. Please save me and solve!
\(\begin{array}{|r|rcr|} \hline n & n\cdot(n+1)\cdot(n+2) \\ \hline 0 & 0\cdot 1 \cdot 2 &=& 0 \\ 1 & 1\cdot 2 \cdot 3 &=& 6 \\ 2 & 2\cdot 3 \cdot 4 &=& 24 \\ 3 & 3\cdot 4 \cdot 5 &=& 60 \\ 4 & 4\cdot 5 \cdot 6 &=& 120 \\ 5 & 5\cdot 6 \cdot 7 &=& 210 \\ 6 & 6\cdot 7 \cdot 8 &=& 336 \\ 7 & 7\cdot 8 \cdot 9 &=& 504 \\ 8 & 8\cdot 9 \cdot 10 &=& 720 \\ \cdots & \cdots \\ \hline \end{array}\)
see: https://en.wikipedia.org/wiki/Tetrahedral_number
0,6,24,60,120,210? Yes!. There is a pattern, which is: n^3 - n. So, your next terms will be:
336, 504, 720.....etc.
0, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969.......etc.
There is a pattern generated by this rather complex formula:
G_n(a_n)(z) = (z (-z^3+4 z^2-6 z+4))/(z-1)^4
There may be simpler generating formulas, but can't see it!.
+26393 Is there a pattern to 0,6,24,60,120,210? This is fiendishly hard. Or how about 0,4,10,20,35. Please save me and solve!
\(\begin{array}{|r|rcr|} \hline n & n\cdot(n+1)\cdot(n+2) \\ \hline 0 & 0\cdot 1 \cdot 2 &=& 0 \\ 1 & 1\cdot 2 \cdot 3 &=& 6 \\ 2 & 2\cdot 3 \cdot 4 &=& 24 \\ 3 & 3\cdot 4 \cdot 5 &=& 60 \\ 4 & 4\cdot 5 \cdot 6 &=& 120 \\ 5 & 5\cdot 6 \cdot 7 &=& 210 \\ 6 & 6\cdot 7 \cdot 8 &=& 336 \\ 7 & 7\cdot 8 \cdot 9 &=& 504 \\ 8 & 8\cdot 9 \cdot 10 &=& 720 \\ \cdots & \cdots \\ \hline \end{array}\)
see: https://en.wikipedia.org/wiki/Tetrahedral_number
+2511 Amazing! Or Amazing BS!
Guest #4 recognized the formula (g(n) = n3 - n) as a solution for generating these numbers (0, 6, 24, 60, 120, 210). (Heureka’s expansion makes this very clear.)
However, the guest totally missed that this same formula, when divided by 6 ((g(n) =n3- n)/6), generates the solution for the second set of numbers) (0, 1, 4, 10, 20, 35) –maybe because the asker left out the (1) in the series. He did pluck this convoluted mess from one of Hade’s rivers: G_n(a_n)(z) = (z (-z^3+4 z^2-6 z+4))/(z-1)^4.
After hours of playing with it, I’ve not been able to make it work, either because my skills are poor –or because it really doesn’t work.
Can anyone explain why this formula will or will not work for either of these series (0, 1, 4, 10, 20, 35) or (0, 4, 10, 20, 35)
Congratulation GingerAle!!. You appear to have stumbled upon a very simple formula for generating the 2nd series!. I don't quite understand your modesty when you say "After hours of playing with it, I’ve not been able to make it work, either because my skills are poor –or because it really doesn’t work."
The 35th term of the 1st. one is: 35^3 - 35 =42,840
The 35th term of the 2nd. one is: 42,840/6=7,140. By using the formula provided by "Guest #4", "from one of Hade’s rivers: G_n(a_n)(z) = (z (-z^3+4 z^2-6 z+4))/(z-1)^4", I derive the 35th term of the this series as: 7,140!!!!!. I tried a couples more terms for both series and your suggestion for the 2nd series continues to hold true! Success!!.
+2511 Someone is mixing up the metaphors here.
It’s not “my formula” that I played with for hours. That formula popped out like an oversized “Where’s Wally” character. Heureaka probably didn’t mention it because that would be like saying, “Notice the big elephant in the room.”
What I played with for hours was the formula of “little mice” waltzing in the shadows. This one: G_n(a_n)(z) = (z (-z^3+4 z^2-6 z+4))/(z-1)^4. The Rube Goldberg formula (but only if it works).
I do not know how to make these series (0, 1, 4, 10, 20, 35) or (0, 4, 10, 20, 35) appear when [n=0, 1, 2, 3, 4, 5, . . . ] probably because I do not know how to do [nested sub-scripted indexes??] Gn(an)(z)=Rube Goldberg formula. I’m assuming the formula works but is poorly explained – z cannot = 1 else there is a division by zero.
By using the formula provided by "Guest #4", "from one of Hade’s rivers: G_n(a_n)(z) = (z (-z^3+4 z^2-6 z+4))/(z-1)^4", I derive the 35th term of the this series as: 7,140!!!!!. I tried a couples more terms for both.
This is great! Instead of just saying you did it, can you show an explicit example? step by step in time. Remember a waltz is in ¾ time not 35
We genetically enhanced chimps are happy to learn from choreographed, waltzing mice, plucked, along with the formula, from one of Hade’s rivers –even if they do step a little out of time, but if they are just running around looking for the big cheese, we’re going to snack on them. Right now, they are looking very tasty.
I think it is " back to school for you, and the study of 'complex functions'"!!.
Calm down guys! The "Hade's Formula" holds true, for it generates the following series:
0, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180, 16215, 17296, 18424, 19600, 20825, 22100, 23426, 24804, 26235, 27720, 29260, 30856, 32509, 34220, 35990, 37820, 39711, 41664, 43680, 45760, 47905, 50116, 52394, 54740, ........etc.
+2511 Ah ha!!
So the mice really are waltzing only to grace notes. (Grace notes are independent of the time signature.)
Yep, I do need to go back to math class. Still, an explicit example would go a long way in knowing which chapter to study. Any baboon can type numbers into a computer and it impresses other baboons. Pressing a button on an electronic piano is great; playing it yourself means, you didn’t learn this from the parrot in the tree next to yours.
