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.

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

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