There is a pattern following, and trying to find the algebraic expression Each layer (from the top). Diagram.  So the first layer has 1, second has 4, third has 9, and the fourth has 16. That's how the sequence is increasing. What I'm looking for is, When the second layer is added with the first layer, Third layer is added with the second and first, Fourth is added with third,second and first. So something like this.   I am trying to find the algebraic expression for this pattern. Any ideas?? Thank you
6333
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:  $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$  I really have no idea why this statement is true. Can someone please explain why this is true and if possible show how to arrive at one given the other?
118084
Hello,      I've gotten far on this exercise, with the following insight:   Here is a matrix of examples (vertical-axis is n=1,2,3,4,5,6,7,8; horizontal-axis is k=1,2,3,4)  1: 1  1   1   1     2: 2  3   4   5     3: 3  6  10  15     4: 4 10  20  35     5: 5 15  35  70     6: 6 21  56 126     7: 7 28  84 210     8: 8 36 120 330   Now, there is an obvious pattern among the numbers, being triangular numbers, sum of triangular numbers, sum of sum of triangular numbers, and so on. My question is:  Can you help me find closed form expressions for the sums of the entries:   Down the columns Across the rows And for general n,k?   Thanks!
25667
DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic.  QUESTION: What is the significance of starting the Fibonacci sequence with $0,1$ ?  For instance, if I picked any two random integers, say 2 and 7, to start a sequence would I actually be creating some multiple or derivation of the Fibonacci sequence?  Is there a general mathematical explanation for the relationship between any sequence represented by $a[0] = x, a[1] = y, a[n] = a[n-1] + a[n-2]$ and the Fibonacci sequence?  Or, back to my example sequence, is there a general mathematical relationship between:  $2,7,9,16,25,41,66,107,173,280...$  and  $0,1,1,2,3,5,8,13,21,34...$  Perhaps the Golden Ratio explains it somehow?  Any help would be appreciated.
78087
I am trying to do this question but don't know where to go from here: The question: For $n\ge1$ let $t_n$ be the number of ways to tile the squares of a 2xn checkerboard using 1x2(which can be rotated to be 2x1 tiles) and 2x2 tiles, if the 1x2 tiles come in 5 different colors and the 2x2 tiles in 4 different colors. Find and solve a recurrence relation for $t_n$.  I think I first part of 1x2 tiles to be : $t_n=3t_{n-1} +9t_{n-2}$  Is that part right? and how do i go from here
10752
Are the numbers appearing as coefficients in the following sequence of polynomials known? Is there a known recurrence relation to compute them?  \begin{align*}   (x+y) &amp;= x+y \\   (x+y)(x+2y) &amp;= x^2+3yx+2y^2 \\   (x+y)(x+2y)(x+3y) &amp;= x^3+6yx^2+11y^2x+6y^3 \\   (x+y)(x+2y)(x+3y)(x+4y) &amp;= x^4+10yx^3+35y^2x^2+50y^3x+24y^4 \\  &amp;\text{etc.} \end{align*}
39841
How to use the  method?  Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$,  where $x_i\in\mathbb{N}$.  Is this the correct time to apply the method?  This is being repurposed in an effort to cut down on duplicates, see here:     and here:
25377
I did the following base case $n = 5$ $$\begin{align*}     4(5) &amp;\lt 2 ^5\\     20 &amp;\lt 32 \end{align*}$$ So true.  $$\begin{align*}     4n &amp;\lt 2^n\\ n &amp;\lt 2^{n-2}\\ \log_2(n)+2 &amp;\lt n \end{align*}$$ But I don't think this is right. Where do I add in the $(n+1)$.
75265
This is not for homework, but I am not great at counting arguments and would like some feedback.  The question asks     Let $n \in \mathbb{N}$. How many $\alpha \in S_n$ are there such that $\alpha^2 = 1$?   I know that, if $\alpha^2 = 1$, then either $\alpha = 1$ or $\alpha$ is the product of disjoint transpositions.  If $\alpha = (i, j)$ is a single transposition, then there are $\frac{1}{2^1 \cdot 1!} \binom{n}{2}$ such $\alpha$ (the $2^1$ and $1!$ are put in the denominator to help in noticing the pattern later).  If $\alpha = (i, j)(k, l)$ is the product of $2$ disjoint transpositions, then there are $\frac{1}{2^2 \cdot 2!} \binom{n}{2} \binom{n-2}{2}$ such $\alpha$, where the $2^2$ appears in the denominator to account for the cyclic permutations of each transposition, and the $2!$ appears to account for the permutation of the transpositions themselves.  If $\alpha$ is the product of $3$ disjoint transpositions, then there are $\frac{1}{2^3 \cdot 3!} \binom{n}{2} \binom{n-2}{2} \binom{n-4}{2}$ such $\alpha$.  Extrapolating from this, I find that the total number of $\alpha \in S_n$ such that $\alpha^2 = 1$ is $$ 1 + \sum_{i=1}^{\lfloor \frac{n}{2} \rfloor} \frac{1}{2^i \cdot i!} \prod_{k=0}^{i-1} \binom{n-2k}{2}. $$  Does this look OK?  It looks like a rather ugly answer to me, so I have my doubts.  Any input would be welcomed.
87954
Let  $$A_2 = \left[ \begin{array}{cc} 0 &amp; 1\\ 1 &amp; 0 \end{array}\right]$$  $$A_3 = \left[ \begin{array}{ccc} 0 &amp; 1 &amp; 1\\ 1 &amp; 0 &amp; 1 \\ 1 &amp; 1 &amp; 0 \end{array}\right]$$  $$A_4 = \left[ \begin{array}{cccc} 0 &amp; 1 &amp; 1 &amp; 1\\ 1 &amp; 0 &amp; 1 &amp; 1 \\ 1 &amp; 1 &amp; 0 &amp; 1 \\ 1 &amp; 1 &amp; 1 &amp; 0\end{array}\right]$$  and so on for $A_n$.  I was asked to calculate the determinant for $A_1, A_2, A_3, A_4$ and then guess about the determinant for $A_n$ in general. Of course the pattern is clear that  $$ \det A_n = (n-1)(-1)^{n-1} $$  but I was wondering as to what the proof of this is. I tried to be clever with cofactor expansions but I couldn't get anywhere.  Could someone explain it to me please?
19271
When showing that $\mathbb{N}\times\mathbb{N}$ is in bijection with $\mathbb{N}$, it seems standard to give a proof by picture that shows a way to systematically weave through all the points in $\mathbb{N}\times\mathbb{N}$ and label each one as you go.    I know there is a polynomial expression for this method, given by $$ J(m,n)=[1+2+\cdots+(m+n)]+m=\frac{1}{2}[(m+n)^2+3m+n] $$ where $m$ is the usual $x$-coordinate and $n$ the usual $y$-coordinate.  But how does one "see" how this formula is arrived at? I know how to manipulate the middle expression to arrive at the rightmost expression, but how does the middle expression relate to the weaving pattern through $\mathbb{N}\times\mathbb{N}$? Thank you.
31379
Consider the sequence defined by $$ \begin{cases} a_0=1\\ a_1=2\\ a_n=3a_{n-1}+4a_{n-2} &amp; \text{if }n\ge 2 \end{cases} .$$ Find a closed form for $a_n$.    I tried listing out examples, but I don't see any common pattern between them.  All solutions are greatly appreciated.
96642
What does the number 2 mean in this picture?    Thanks.   Sorry I'm not a math expert.  I need this to make a program in C to generate a trinomial triangle for a guy who asked it on StackOverflow (it is his homework). Gotta earn that reputation. You know how it is!
104670
I'm looking for a closed-form expression (in terms of $n$), that will give the sequence  $$ (s_n) = \left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right). $$  Can anyone think of one? I made a related post to this question several minutes ago but I realized I was interpreting the sequence wrong.
66935
Okay, so I mostly worked this out, and I even created lattice diagrams as shown below. But I have a specific question about finding intermediate fields, which I will ask shortly. Let $\alpha = \sqrt[4]{2}$ and $\omega = e^{\frac{\pi}{4}i} = i$.  Then $L = \mathbb{Q}(\alpha, i)$ is the splitting field of $x^4 -2$ over $\mathbb{Q}$. Also, the Galois group $\Gamma_\mathbb{Q}(x^4 - 2) = D_8$ acts on the roots $\alpha, \alpha i, -\alpha,$ and $-\alpha i$, and is generated by rotation $\sigma$ and reflection $\tau$, where $\sigma(i) = i, \sigma(\alpha) = \alpha i$ and $\tau(\alpha) = \alpha, \tau(i) = -i$. To find the intermediate fields between $L$ and $\mathbb{Q}$, find the subgroups of $D_8$ instead with the idea that finding subgroups is easier and better understood than finding intermediate fields. Then from the subgroups, use the Galois correspondence to get all the intermediate fields. There are 10 subgroups of $D_8$ which must correspond to 10 intermediate fields.  Well, I pieced together 8 obvious candidates for intermediate fields, and in the end, I had to look up the other 2 which were $\mathbb{Q}(\alpha(1 + i))$ and $\mathbb{Q}(\alpha(1 - i))$. Those two seemed strange until I realized that $\sqrt{8\alpha^2 i} = \alpha(1 + i)$. Finally, I was able to check fixed fields to verify the exact correspondence, and come up with the diagrams. Question: Is there a systematic approach to finding and connecting up the corresponding intermediate fields once all the subgroups are known? I'm guessing, in general and maybe in this example with $D_8$, there isn't a good, canonical way to anticipate and construct the field extensions?  The structure of groups and subgroups, as stated earlier, is easier and better understood than the structure of field extensions. Maybe this makes sense because the groups are finite and have only one operation, and fields are often infinite and have two operations.  and  UPDATE: In , Richard Borcherds clearly describes how to obtain the two nonobvious intermediate fields from the subgroups. Specifically, add the roots $\alpha$ and $\alpha i$ to fix by reflection one way, and then add the roots $\alpha$ and $-\alpha i$ to fix by reflection the other way.
66047
Question: Why is the table of $GF(4)$ look like the one below? I know it has to do with the fact that 4 is composite Let $GF(4) = \{0,1,B,D\}$ Addition: $$ \begin{array}{c|cccc} +  &amp; 0&amp;   1&amp;   B &amp;  D  \\ \hline     0&amp;   0 &amp;  1 &amp;  B &amp;  D \\  1 &amp;  1 &amp;  0 &amp;  D &amp;  B  \\ B &amp;  B &amp;  D &amp;  0 &amp;  1  \\ D &amp;  D &amp;  B &amp;  1 &amp;  0  \end{array} $$ Multiplication: $$\begin{array}{c|cccc} \cdot &amp;   0 &amp;  1 &amp;  B &amp;  D  \\ \hline 0 &amp;  0 &amp;  0 &amp;  0 &amp;  0  \\ 1 &amp;  0 &amp;  1 &amp;  B &amp;  D  \\ B &amp;  0 &amp;  B &amp;  D &amp;  1  \\ D &amp;  0 &amp;  D &amp;  1 &amp;  B     \end{array}$$
99833
Currently working on this problem and I'm having trouble seeing how it goes from one line to the next.    $45x \equiv 63\mod 11$ goes to $x \equiv 8\mod 11$     Any help would be awesome thanks.  Please try to explain how it works to help me see thanks.
123656
We are given ($a_1,a_2,....,a_m$) and ($b_1, b_2,....,b_n$) sequences with non-negative integers.   Decide whether it's possible and if it is construct a matrix $\Re^{m x n}$ of ones("1") and zeros("0") where the number of ones("1") in row $i$ is $\forall i $ : $a_i$ and in column $j$ is $\forall j$ : $b_j$   Any hint is greatly appreciated.