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A265400
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a(n) = one-based index to the nearest horizontally or vertically adjacent inner neighbor in square-grid spirals, or 0 if n is one of the corner cases A033638.
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3
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0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 3, 0, 3, 4, 5, 0, 5, 6, 7, 0, 7, 8, 9, 10, 0, 10, 11, 12, 13, 0, 13, 14, 15, 16, 17, 0, 17, 18, 19, 20, 21, 0, 21, 22, 23, 24, 25, 26, 0, 26, 27, 28, 29, 30, 31, 0, 31, 32, 33, 34, 35, 36, 37, 0, 37, 38, 39, 40, 41, 42, 43, 0, 43, 44, 45, 46, 47, 48, 49, 50, 0, 50, 51, 52, 53, 54, 55, 56, 57, 0, 57, 58, 59, 60, 61, 62, 63, 64, 65, 0
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OFFSET
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1,9
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COMMENTS
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By convention we set a(1) = 0, because as 1 is a starting point of such spirals, it has no "inner neighbors" for itself.
This sequence is useful when constructing spiral-based sequences like A260643.
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LINKS
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FORMULA
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a(1) = a(2) = 0. If 3 <= n <= 8, then a(n) = 1 - (n mod 2), and for n >= 8, if A240025(n-1) is not zero [when n is in A033638], then a(n) = 0, otherwise, if A240025(n-2) is not zero [when n is one more than some term of A033638], then a(n) = A033638(A000267(n)-4), otherwise, a(n) = 1 + a(n-1).
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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 and C for 12). The first square corresponds with n, and the second square with the value of a(n):
05430
543C 50103
612B 61002
789A 70120
0789A0
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For each n > 1, 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 1 for 4, 6 and 8, while it is 2 for 9 and 11 and 3 for 12] unless no such additional neighbor exists, in which case a(n) = 0 (this occurs when n is one of the A033638, Quarter-squares plus 1).
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PROG
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(Scheme, with memoizing-macro definec, demonstrating the given recursive formula)
;; Another version which uses the variant A265410:
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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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