

A063826


Let 1, 2, 3, 4 represent moves to the right, down, left and up; this sequence describes the movements in the clockwise square spiral (a.k.a. Ulam Spiral).


34



1, 2, 3, 3, 4, 4, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4
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OFFSET

0,2


COMMENTS

Sequence starts with 1, 2, 3, then can be broken into groups of 8n+4 members, so if n is incremented, starting at 1, the groups follow the following pattern: 3 occurs at the beginning of the group, 4 then occurs 2n times, 1 occurs 2n+1 times, 2 occurs 2n+1 times, 3 occurs 2n+1 times; so each group has 8n+4 terms.
Simpler description: Groups of 2*(2n1) + 2*(2n) = 8n  2 terms, n = 1, 2, 3, ..., consisting of 2n1 times 1, then 2n1 times 2; then 2n times 3, then 2n times 4. The nth group starts at index (4n  6)n + 2 and ends at index (4n + 2)n  1.  M. F. Hasler, Aug 08 2020


LINKS

Harry J. Smith, Table of n, a(n) for n = 0..1000
D. Alpern, Ulam's Spiral
Adrian J. F. Leatherland, The mysterious Ulam spiral phenomenon


FORMULA

1 <= a(n) <= 4 with a(n) == floor(sqrt(4n + 1)) (mod 4).  M. F. Hasler, Aug 08 2020


EXAMPLE

Breaking into the groups, we have: 1, 2, 3 n=1: 3, 4, 4, 1, 1, 1, 2, 2, 2, 3, 3, 3, n=2: 3, 4, 4, 4, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3 n=3: 3, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3 and so on.
The spiral of numbers which shows in which order the steps in direction right, down, left and up are made, is depicted for example in sequence A174344.  M. F. Hasler, Aug 08 2020


MATHEMATICA

a[n_] := Mod[Floor[Sqrt[4*n + 1]] + 3, 4] + 1; Table[a[n], {n, 0, 104}] (* JeanFrançois Alcover, Nov 30 2016 adapted from PARI *)
Join[{1, 2, 3}, Flatten[Table[{{3}, PadRight[{}, 2n, 4], Table[PadRight[{}, 2n+1, k], {k, 3}]}, {n, 5}]]] (* Harvey P. Dale, Jun 29 2019 *)


PROG

(PARI) A063826(n)=(sqrtint(4*n+1)+3)%4+1 \\ To see the terms: apply(A063826, [0..99])


CROSSREFS

Cf. A000267, A174344.
Sequence in context: A096827 A298321 A226142 * A320120 A152983 A331377
Adjacent sequences: A063823 A063824 A063825 * A063827 A063828 A063829


KEYWORD

easy,nice,nonn


AUTHOR

Wai Ha Lee (Wainson(AT)hotmail.com), Aug 20 2001


STATUS

approved



