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A214526 Manhattan distances between n and 1 in a square spiral with positive integers and 1 at the center. 24
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Spiral begins:

  49  26--27--28--29--30--31

   |   |                   |

  48  25  10--11--12--13  32

   |   |   |           |   |

  47  24   9   2---3  14  33

   |   |   |   |   |   |   |

  46  23   8   1   4  15  34

   |   |   |       |   |   |

  45  22   7---6---5  16  35

   |   |               |   |

  44  21--20--19--18--17  36

   |                       |

  43--42--41--40--39--38--37

LINKS

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

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A175835 A123738 A194511 * A245038 A161312 A161246

Adjacent sequences:  A214523 A214524 A214525 * A214527 A214528 A214529

KEYWORD

nonn,easy

AUTHOR

Alex Ratushnyak, Aug 08 2012

STATUS

approved

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Last modified May 26 13:17 EDT 2022. Contains 354092 sequences. (Running on oeis4.)