superincreasing sequence

A sequence $\\{s_{j}\\}$ of real numbers is superincreasing if $\\displaystyle s_{n+1}>\\sum_{j=1}^{n}s_{j}$ for every positive integer $n$ . That is, any element of the sequence is greater than all of the previous elements added together.

A commonly used superincreasing sequence is that of powers of two ( $s_{n}=2^{n}$ .)

Suppose that $\\displaystyle x=\\sum_{j=1}^{n}a_{j}s_{j}$ . If $\\{s_{j}\\}$ is a superincreasing sequence and every $a_{j}\\in\\{0,1\\}$ , then we can always determine the $a_{j}$ ’s simply by knowing $x$ . This is analogous to the fact that, for any natural number, we can always determine which bits are on and off in the binary bitstring representing the number.

单词	SuperincreasingSequence
释义	superincreasing sequence A sequence $\\{s_{j}\\}$ of real numbers is superincreasing if $\\displaystyle s_{n+1}>\\sum_{j=1}^{n}s_{j}$ for every positive integer $n$ . That is, any element of the sequence is greater than all of the previous elements added together. A commonly used superincreasing sequence is that of powers of two ( $s_{n}=2^{n}$ .) Suppose that $\\displaystyle x=\\sum_{j=1}^{n}a_{j}s_{j}$ . If $\\{s_{j}\\}$ is a superincreasing sequence and every $a_{j}\\in\\{0,1\\}$ , then we can always determine the $a_{j}$ ’s simply by knowing $x$ . This is analogous to the fact that, for any natural number, we can always determine which bits are on and off in the binary bitstring representing the number.
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