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 A265410 a(n) = one-based index to the nearest horizontally or vertically adjacent inner neighbor in square-grid spirals, and to the nearest diagonally adjacent inner neighbor when n is one of the corner cases A033638. 6
 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 10, 10, 10, 11, 12, 13, 13, 13, 14, 15, 16, 17, 17, 17, 18, 19, 20, 21, 21, 21, 22, 23, 24, 25, 26, 26, 26, 27, 28, 29, 30, 31, 31, 31, 32, 33, 34, 35, 36, 37, 37, 37, 38, 39, 40, 41, 42, 43, 43, 43, 44, 45, 46, 47, 48, 49, 50, 50, 50, 51, 52, 53, 54, 55, 56, 57, 57, 57, 58, 59, 60, 61, 62, 63, 64, 65, 65 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS By convention: a(1) = 0 because as 1 is a starting point of such spirals, it has no "inner neighbors" for itself. Each n occurs A265411(n) times. LINKS Antti Karttunen, Table of n, a(n) for n = 1..10001 Antti Karttunen, R6RS-Scheme program for computing this sequence FORMULA a(1) = 0; for 1 < n < 8, a(n) = 1 and for n >= 8: if either A240025(n-1) or A240025(n-2) is not zero [when n or n-1 is in A033638], then a(n) = a(n-1), otherwise, a(n) = 1 + a(n-1). a(1) = 0; for 1 < n < 8, a(n) = 1 and for n >= 8: a(n) = a(n-1) + (1-A240025(n-1))*(1-A240025(n-2)). [The same formula in a more compact form.] Other identities. For all n >= 0: a(A265413(n)) = n. [Sequence is the least monotonic left inverse of A265413.] a(A265412(n)) = n. [Also inverse of A265412.] EXAMPLE 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):                     55433             543C    51113C             612B    61012B             789A    71122A                     7789AA - 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 n is in the corner (one of the terms of A033638), in which case the value is the nearest diagonally adjacent neighbor towards the center. PROG (Scheme, with memoizing-macro definec) (definec (A265410 n) (cond ((= 1 n) 0) ((<= n 7) 1) ((or (= 1 (A240025 (- n 1))) (= 1 (A240025 (- n 2)))) (A265410 (- n 1))) (else (+ 1 (A265410 (- n 1)))))) CROSSREFS One more than A265409(n-1). Cf. A000267, A033638, A240025. Cf. A265400 (a variant). Cf. A265411, A265412, A265413 (positions of records, i.e., where value increases). Sequence in context: A219502 A219704 A253144 * A029249 A025770 A342098 Adjacent sequences:  A265407 A265408 A265409 * A265411 A265412 A265413 KEYWORD nonn AUTHOR Antti Karttunen, Dec 09 2015 STATUS approved

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Last modified September 21 12:28 EDT 2021. Contains 347598 sequences. (Running on oeis4.)