A396877 a(n) is the first step at which the n-th lattice point in counterclockwise square-spiral order becomes an endpoint in a growth process of alternating perpendicular bars starting from one horizontal bar.

4, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11

Offset

0,1

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Seiichi Manyama, step 8

Formula

a(n^2) = n for n > 0.

a(0) = 4. For n > 0, let k = ceiling((sqrt(n+1)-1)/2), m = (n - 4*k*(k-1) - 1) mod 4*k. a(n) = 2*k - 1 + (m mod 2) + 2*[m >= 2*k and m even].

Example

Beginning of the square-spiral arrangement.
  4---5---4---5---4   .
  |               |   .
  3   2---3---2   3   .
  |   |       |   |   .
  4   1   4---1   4   .
  |   |           |   .
  3   2---3---2---3   .
  |                   .
  4---5---4---5---4---5

Mathematica

A396877[n_] := If[n == 0, 4, With[{k = Ceiling[(Sqrt[n+1] - 1)/2]}, {m = Mod[n - 4*k*(k-1) - 1, 4*k]}, 2*k - 1 + Mod[m, 2] + If[m >= 2*k && EvenQ[m], 2, 0]]];
Array[A396877, 100, 0] (* Paolo Xausa, Jun 09 2026 *)

Prog

(PARI) a(n) = if(n==0, 4, my(k=ceil((sqrt(n+1)-1)/2), m=(n-4*k*(k-1)-1)%(4*k)); 2*k-1+m%2+2*(m>=2*k)*(1-m%2));

Crossrefs

Sequence in context: A176041 A131100 A378626 * A321091 A375576 A064124

Adjacent sequences: A396874 A396875 A396876 * A396878 A396879 A396880

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 09 2026

Status

approved