

A265410


a(n) = onebased index to the nearest horizontally or vertically adjacent inner neighbor in squaregrid 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
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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, R6RSScheme program for computing this sequence


FORMULA

a(1) = 0; for 1 < n < 8, a(n) = 1 and for n >= 8: if either A240025(n1) or A240025(n2) is not zero [when n or n1 is in A033638], then a(n) = a(n1), otherwise, a(n) = 1 + a(n1).
a(1) = 0; for 1 < n < 8, a(n) = 1 and for n >= 8: a(n) = a(n1) + (1A240025(n1))*(1A240025(n2)). [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 n1, 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 memoizingmacro 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(n1).
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 A283875
Adjacent sequences: A265407 A265408 A265409 * A265411 A265412 A265413


KEYWORD

nonn


AUTHOR

Antti Karttunen, Dec 09 2015


STATUS

approved



