Template:Sequence of the Day for October 11

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

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A001020:

11 n, 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

.