A265400 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.

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

Offset: 1

Author

Antti Karttunen, Dec 09 2015

Keywords

• nonn

Comments

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.

Examples

			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
-
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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10001
• Antti Karttunen, R6RS-Scheme program for computing this sequence

Crossrefs

Cf. A000035, A000267, A033638 (positions of zeros), A240025, A260643.

Cf. A265410 (a variant).

Formula

If A240025(n-1) = 1 [when n is in A033638], then a(n) = 0, otherwise a(n) = A265410(n).

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).