Question

Write a simple formula that generates the below mentioned terms.

a)  1,2,2,3,4,4,5,6,6,7,8,8,...

b)   1,10,11,100,101,110,111,1000,1001,1010,1011

Answer 1

a.) 1,2,2,3,4,4,5,6,6,7,8,8,....

General formula $a(n)=a(n-1)+a(n-3)-a(n-4)\ \ \ \text{For n>4}$

b.) 1,10,11,100,101,110,111,1000,1001,1010,1011

Then if n is a positive integer, n can be expressed uniquely in the form

Given a number n all you need to do it find the k where $2^1+2^2+2^3 ...+2^k < n < 2^1+2^2+2^3 ...2^{(k+1)}$ .

The binary representation of $n-(2^1+2^2+2^3 ...2^k)$ in k+1 bits is the answer.