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%I #12 Dec 18 2015 11:25:48
%S 0,0,0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,5,6,6,6,7,8,9,9,9,10,11,12,12,12,
%T 13,14,15,16,16,16,17,18,19,20,20,20,21,22,23,24,25,25,25,26,27,28,29,
%U 30,30,30,31,32,33,34,35,36,36,36,37,38,39,40,41,42,42,42,43,44,45,46,47,48,49,49,49,50
%N a(n) = index to the nearest inner neighbor in Ulam-style square-spirals using zero-based indexing.
%C Each n occurs A265411(n+1) times.
%C Useful when defining recurrences like A078510 and A265408.
%H Antti Karttunen, <a href="/A265409/b265409.txt">Table of n, a(n) for n = 1..10000</a>
%F If n <= 7, a(n) = 0 for n >= 8: if either A240025(n) or A240025(n-1) is not zero [when n or n-1 is in A002620], then a(n) = a(n-1), otherwise, a(n) = 1 + a(n-1).
%F If n <= 7, a(n) = 0, for n >= 8, a(n) = a(n-1) + (1-A240025(n))*(1-A240025(n-1)). [The same formula in a more compact form.]
%F a(n) = A265410(n+1) - 1.
%F Other identities. For all n >= 0:
%F a(n) = n - A265359(n).
%e We arrange natural numbers as a counterclockwise spiral into the square grid in the following manner (here A stands for 10, B for 11). The first square corresponds with n (where the initial term 0 is at the center), and the second square with the value of a(n). This sequence doesn't specify a(0), thus it is shown as an asterisk (*):
%e 44322
%e 432B 40002B
%e 501A 50*01A
%e 6789 600119
%e 667899
%e -
%e For each n > 0, 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 0 for 3, 5 and 7, while it is 1 for 8, 9 and A (10) and 2 for B (11)] unless n is in the corner (one of the terms of A002620), in which case the value is the nearest diagonally adjacent neighbor towards the center, e.g. 0 for 2, 4 and 6, while it is 1 for 9).
%e See also the illustration at A078510.
%o (Scheme, with memoization macro definec)
%o (definec (A265409 n) (cond ((<= n 7) 0) ((or (= 1 (A240025 n)) (= 1 (A240025 (- n 1)))) (A265409 (- n 1))) (else (+ 1 (A265409 (- n 1))))))
%Y One less than A265410(n+1).
%Y Cf. A002620, A240025, A265359.
%Y Cf. also A033951, A063826, A078510, A265408.
%K nonn
%O 1,11
%A _Antti Karttunen_, Dec 13 2015
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