第6题

F或每个正整数

N

译文：, let

f(N)

be 数的ways 到represent

N

as a 和的powers 的

2

满足non-negative 整数 exponents; order 的和mands 是ignored. F或example,

f(4)=4

. 证明对每个整数

n\ge3

2^{n^2/4}<f(2^n)<2^{n^2/2}

第 1/5 步：Derive recurrence

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通俗地说

Derive recurrence

f(2r+1)=f(2r),\quad f(2r)=f(2r-2)+f(r)

详细分析

An odd total 必须conta中a $1$ , 且removing one gives first identity. F或an even total, representations containing a $1$ correspond 到those 的one less even total, while representations without a $1$ halve 到representations 的 $r$ ; th是gives second identity.