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For every positive integer , there is a unique
finite sequence of distinct nonconsecutive (not necessarily positive) integers , ..., such that
where is the golden
ratio.
For example, for the first few positive integers,
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(OEIS A104605).
The numbers of terms needed to represent for , 2, ... are
given by 1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 4, ... (OEIS A055778),
which are also the numbers of 1s in the base- representation
of .
The following tables summarizes the values of that require exactly
powers of in their representations.
| OEIS | numbers requiring exactly powers | | 2 | A005248 | 2, 3, 7, 18, 47, 123, 322, 843, ... | | 3 | A104626 | 4,
5, 6, 8, 19, 48, 124, 323, 844, ... | | 4 | A104627 | 9,
10, 12, 13, 14, 16, 17, 20, 21, 25, ... | | 5 | A104628 | 11,
15, 22, 23, 24, 26, 30, 31, 32, 34, ... |
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