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A363665
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Starting with a(1) = 1, the lexicographically earliest sequence of integers on a square spiral such that every number equals the sum of its eight adjacent neighbors.
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1
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1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 1, -2, 0, 0, 0, -1, 0, 0, 0, -2, 2, 0, 0, -1, 2, -3, 0, 0, 3, -2, 2, -3, 0, 0, 3, -1, -1, 1, 0, 0, 1, 0, 0, -5, 8, 0, 0, -5, 4, 0, 4, -7, 0, 0, 6, -6, 4, -4, 8, -7, 0, 0, 2, -2, 4, -4, 0, 7, 0, 0, -8, 6, 3, -8, 10, -15, 16, 0, 0, -9, 6, -5, 7, -8, 13
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OFFSET
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1,12
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COMMENTS
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As the terms are not distinct the first two numbers of any new row or column will always be zero. In the first 500000 terms the last zero that is not at the beginning of a row or column is a(190) = 0. Is it unknown if more such zeros exist. In the same range the smallest positive numbers not yet occurring are 5, 9, 11, 12, 15, 19, 20, ... . It is unknown if all integers eventually appear. The terms increase rapidly in size; in the first 500000 terms the largest positive term is a(499848) = 1267...5398, a number with 226 digits.
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LINKS
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EXAMPLE
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The spiral begins:
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0__-3___2__-2___3___0___0 -7
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0 0__-2___1___0___0 -3 4
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3 0 0___0___0 -2 2 0
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-1 0 0 1___0 0 -1 4
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-1 -1 0___0___1___0 0 -5
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1 0___0___0__-2___2___0 0
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0___0___1___0___0__-5___8___0
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a(9) = 1 as a(1) = 1 and a(2)..a(8) = 0, therefore a(9) = 1 so the sum of the eight numbers around a(1) equals 1.
a(12) = -2 as a(2) = 0 while a(1), a(9) = 1, a(2)..a(4), a(8), a(10), a(11) = 0, therefore a(12) = -2 so the sum of the eight numbers around a(1) equals 0.
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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