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

1,9

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10001

Antti Karttunen, R6RS-Scheme program for computing this sequence

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).

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):
                    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).

Prog

(Scheme, with memoizing-macro definec, demonstrating the given recursive formula)
(definec (A265400 n) (cond ((<= n 3) 0) ((<= n 8) (- 1 (A000035 n))) ((= 1 (A240025 (- n 1))) 0) ((= 1 (A240025 (- n 2))) (A033638 (- (A000267 n) 4))) (else (+ 1 (A265400 (- n 1))))))
;; Another version which uses the variant A265410:
(define (A265400 n) (if (= 1 (A240025 (- n 1))) 0 (A265410 n)))

Crossrefs

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

Cf. A265410 (a variant).

Sequence in context: A160706 A087509 A274097 * A181871 A269591 A262878

Adjacent sequences: A265397 A265398 A265399 * A265401 A265402 A265403

Keyword

nonn

Author

Antti Karttunen, Dec 09 2015

Status

approved