This site is supported by donations to The OEIS Foundation.

Template:Sequence of the Day for October 11

From OeisWiki
Jump to: navigation, search

Intended for: October 11, 2012

Timetable

  • First draft entered by Alonso del Arte on July 10, 2011
  • Draft reviewed by Mitch Harris on August 9, 2011
  • Draft reviewed by M. F. Hasler on October 10, 2012
  • Draft approved by Daniel Forgues on October 10, 2012, October 8, 2016
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



A001020:
11n, n ≥ 0.
{ 1, 11, 121, 1331, 14641, 161051, ... }

The decimal digits of the powers of 11 give binomial coefficients (which appear in Pascal’s triangle), until some binomial coefficients have more than one digit, where overlapping occurs.

Pascal’s triangle
n

0   1  
1   1 1  
2   1 2 1  
3   1 3 3 1  
4   1 4 6 4 1  
5 1 5 10 10 5 1  

k = 0

1
2
3
4
5  

After 11 4 it doesn’t quite work out, since some binomial coefficients in {1, 5, 10, 10, 5, 1} have more than one decimal digits.

1, 
   5, 
   1  0, 
      1  0, 
            5, 
               1
----------------
1  6  1  0  5  1

In fact, the powers of 11 correspond to a Pascal’s triangle with only one digit per entry, and carry over to the next cell to the left in case of a value   ≥   10.

In general, if
b
appears in Pascal’s triangle in the middle of row
k
, then
(b + 1)n
can be found in Pascal’s triangle written in base
b
up to
(b + 1)k  − 1
.