A214526 Manhattan distances between n and 1 in a square spiral with positive integers and 1 at the center.

0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10

Offset: 1

Author

Alex Ratushnyak, Aug 08 2012

Keywords

• nonn
• easy

Comments

Spiral begins:

49 26--27--28--29--30--31

| | |

48 25 10--11--12--13 32

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47 24 9 2---3 14 33

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46 23 8 1 4 15 34

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45 22 7---6---5 16 35

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44 21--20--19--18--17 36

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43--42--41--40--39--38--37

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10201

Crossrefs

Cf. A137928, A137930, A137931, A002061, A114254, A214176, A214177.

Programs

Mathematica

f[n_] := Block[{o = 2 n - 1, t, w}, t = Table[0, {o}, {o}]; t = ReplacePart[t, {n, n} -> 1]; Do[w = Partition[Range[(2 (# - 1) - 1)^2 + 1, (2 # - 1)^2], 2 (# - 1)] &@ k; Do[t = ReplacePart[t, {(n + k) - (j + 1), n + (k - 1)} -> #[[1, j]]]; t = ReplacePart[t, {n - (k - 1), (n + k) - (j + 1)} -> #[[2, j]]]; t = ReplacePart[t, {(n - k) + (j + 1), n - (k - 1)} -> #[[3, j]]]; t = ReplacePart[t, {n + (k - 1), (n - k) + (j + 1)} -> #[[4, j]]], {j, 2 (k - 1)}] &@ w, {k, 2, n}]; t]; With[{x = Position[#, 1][[1]]}, Table[Total@ Abs[Position[#, n][[1]] - x], {n, Max@ #}]] &@ f@ 6 (* Michael De Vlieger, Feb 16 2018 *)

PARI

a(n) = n--; my(m=sqrtint(n),k=ceil(m/2)); n=abs(n-4*k^2); k+abs(n-if(n>m,3,1)*k); \\ Kevin Ryde, Oct 25 2019

Formula

abs( a(n) - a(n-1) ) = 1.

For n > 1, a(n) = layer(n) + abs(((n-1) mod (2*layer(n)) - layer(n))) (conjectured) where layer(n) = ceiling(0.5*sqrt(n) - 0.5). - Karl R. Stephan, Jan 26 2018

a(n) = abs(A174344(n)) + abs(A274923(n)). - Kevin Ryde, Oct 25 2019