Mathematics desk
< March 31	<< Mar \| April \| May >>	April 2 >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 1[edit]

Adding numbers[edit]

If you have columns of numbers:

#  |  k1 |  k2 |  k3 |  k4 |  k5 |  k6  |
# 0|  1  |  1  |  1  |  1  |  1  |  1  |
# 1|  1  |  2  |  2  |  2  |  2  |  2  |
# 2|  1  |  3  |  4  |  4  |  4  |  4  |
# 3|  1  |  4  |  7  |  8  |  8  |  8  |
# 4|  1  |  5  |  11 |  15 |  16 |  16 |
# 5|  1  |  6  |  16 |  26 |  31 |  32 |
# 6|  1  |  7  |  22 |  42 |  57 |  63 |
# 7|  1  |  8  |  29 |  64 |  99 |  120|
# 8|  1  |  9  |  37 |  93 |  163|  219|
# 9|  1  |  10 |  46 |  130|  256|  382|
  n|  n^0|  n  |(n^2+n+2)/2| (n^3+5n+6)/6|  ?  |  ?  |

Where a number equals to the number below it plus the one on its bottom right. I heard somewhere you add the equations or something. Can you come up with a formula where you input n to get the number for each column? Is there a generalization for all columns? K as the column and n as the row. But most importantly, how do you create equations like this? — Preceding unsigned comment added by Someone with a Question (talk • contribs) 12:48, 1 April 2015 (UTC)[reply]

First of all, your description seems to be upside-down: each cell in your diagram is the sum of cell above it and the cell on its top left; e.g. the 16 in the fifth row is 5 + 11. This sort of thing is called a recurrence relation, though that article has very little to say on multi-variable RRs, which is what your problem is. I found this question on StackExchange which discusses a method of generating a formula for any cell in the grid (the example there is essentially Pascal's triangle, where the rule is that the value of a cell is the sum of the two cells above and to the left of it. (Googling "recurrence relation two variables" found this, and lots of similar hits.) AndrewWTaylor (talk) 14:30, 1 April 2015 (UTC)[reply]

The value in column k of row n is the coefficient of x^(k-1) in the formal power series expansion of

{\frac {(1+x)^{n}}{1-x}}

(assuming column numbering starts at 1). Alternatively, it is the sum of the first k values in the nth row of Pascal's triangle if k <= n, or 2^n otherwise. Gandalf61 (talk) 14:47, 1 April 2015 (UTC)[reply]

I believe this is OEIS: A052509. --RDBury (talk) 16:11, 1 April 2015 (UTC)[reply]