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A265409
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a(n) = index to the nearest inner neighbor in Ulam-style square-spirals using zero-based indexing.
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6
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0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 17, 18, 19, 20, 20, 20, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 29, 30, 30, 30, 31, 32, 33, 34, 35, 36, 36, 36, 37, 38, 39, 40, 41, 42, 42, 42, 43, 44, 45, 46, 47, 48, 49, 49, 49, 50
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OFFSET
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1,11
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COMMENTS
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LINKS
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FORMULA
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If n <= 7, a(n) = 0 for n >= 8: if either A240025(n) or A240025(n-1) is not zero [when n or n-1 is in A002620], then a(n) = a(n-1), otherwise, a(n) = 1 + a(n-1).
If n <= 7, a(n) = 0, for n >= 8, a(n) = a(n-1) + (1-A240025(n))*(1-A240025(n-1)). [The same formula in a more compact form.]
Other identities. For all n >= 0:
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EXAMPLE
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We arrange natural numbers as a counterclockwise spiral into the square grid in the following manner (here A stands for 10, B for 11). The first square corresponds with n (where the initial term 0 is at the center), and the second square with the value of a(n). This sequence doesn't specify a(0), thus it is shown as an asterisk (*):
44322
432B 40002B
501A 50*01A
6789 600119
667899
-
For each n > 0, we look for the nearest horizontally or vertically adjacent neighbor of n towards the center that is not n-1, which will then be value of a(n) [e.g., it is 0 for 3, 5 and 7, while it is 1 for 8, 9 and A (10) and 2 for B (11)] unless n is in the corner (one of the terms of A002620), in which case the value is the nearest diagonally adjacent neighbor towards the center, e.g. 0 for 2, 4 and 6, while it is 1 for 9).
See also the illustration at A078510.
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PROG
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(Scheme, with memoization macro definec)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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