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A063826
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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).
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48
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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
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0,2
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COMMENTS
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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*(2n-1) + 2*(2n) = 8n - 2 terms, n = 1, 2, 3, ..., consisting of 2n-1 times 1, then 2n-1 times 2; then 2n times 3, then 2n times 4. The n-th group starts at index (4n - 6)n + 2 and ends at index (4n + 2)n - 1. - M. F. Hasler, Aug 08 2020
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LINKS
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FORMULA
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1 <= a(n) <= 4 with a(n) == floor(sqrt(4n + 1)) (mod 4). - M. F. Hasler, Aug 08 2020
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EXAMPLE
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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
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MATHEMATICA
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a[n_] := Mod[Floor[Sqrt[4*n + 1]] + 3, 4] + 1; Table[a[n], {n, 0, 104}] (* Jean-François 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 *)
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PROG
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(PARI) A063826(n)=(sqrtint(4*n+1)+3)%4+1 \\ To see the terms: apply(A063826, [0..99])
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Wai Ha Lee (Wainson(AT)hotmail.com), Aug 20 2001
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STATUS
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approved
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