Fractions involving the number of different ways a number can be expressed as a sum of powers of 2
Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice.
For example, f(10)=5 since there are five different ways to express 10:
10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1
It can be shown that for every fraction p/q (p>0, q>0) there exists at least one integer n such that
f(n)/f(n-1)=p/q.
For instance, the smallest n for which f(n)/f(n-1)=13/17 is 241.
The binary expansion of 241 is 11110001.
Reading this binary number from the most significant bit to the least significant bit there are 4 one’s, 3 zeroes and 1 one. We shall call the string 4,3,1 the Shortened Binary Expansion of 241.
Find the Shortened Binary Expansion of the smallest n for which
f(n)/f(n-1)=123456789/987654321.
Give your answer as comma separated integers, without any whitespaces.
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��f(0)=1��f(n)Ϊ��nд��2���ݴεĺ��������ݴγ��ֲ��������εķ�ʽ����
���磬f(10)=5����Ϊ10ǡ����5�ֲ�ͬ�ı�ʾ��ʽ��
10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1
������������p/q����������p>0������q>0�������Ƕ����ҵ�����һ������n��ʹ��
f(n)/f(n-1)=p/q��
���磬ʹ��f(n)/f(n-1)=13/17����С��n��241��
241�Ķ����Ʊ�ʾΪ11110001��
�������Ҷ����������ƴ����ǵõ�4��1��3��0����1��1�����ˣ����dz����ִ�4,3,1��241����ʽ�����Ʊ�ʾ��
�ҳ�������ʽ����С��n�ļ�ʽ�����Ʊ�ʾ��
f(n)/f(n-1)=123456789/987654321.
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