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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.

%I A265400 #24 Dec 13 2015 07:54:42
%S A265400 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,
%T A265400 13,14,15,16,17,0,17,18,19,20,21,0,21,22,23,24,25,26,0,26,27,28,29,30,
%U A265400 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
%N 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.
%C A265400 By convention we set a(1) = 0, because as 1 is a starting point of such spirals, it has no "inner neighbors" for itself.
%C A265400 This sequence is useful when constructing spiral-based sequences like A260643.
%H A265400 Antti Karttunen, <a href="/A265400/b265400.txt">Table of n, a(n) for n = 1..10001</a>
%H A265400 Antti Karttunen, <a href="/A260643/a260643.txt">R6RS-Scheme program for computing this sequence</a>
%F A265400 If A240025(n-1) = 1 [when n is in A033638], then a(n) = 0, otherwise a(n) = A265410(n).
%F A265400 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).
%e A265400 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):
%e A265400                     05430
%e A265400             543C    50103
%e A265400             612B    61002
%e A265400             789A    70120
%e A265400                     0789A0
%e A265400 -
%e A265400 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).
%o A265400 (Scheme, with memoizing-macro definec, demonstrating the given recursive formula)
%o A265400 (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))))))
%o A265400 ;; Another version which uses the variant A265410:
%o A265400 (define (A265400 n) (if (= 1 (A240025 (- n 1))) 0 (A265410 n)))
%Y A265400 Cf. A000035, A000267, A033638 (positions of zeros), A240025, A260643.
%Y A265400 Cf. A265410 (a variant).
%K A265400 nonn
%O A265400 1,9
%A A265400 _Antti Karttunen_, Dec 09 2015