A260643 Start a spiral of numbers on a square grid, with the initial square as a(1) = 1. a(n) is the smallest positive integer not equal to or previously adjacent (horizontally/vertically) to its neighbors. See the Comments section for a more exact definition.

1, 2, 3, 4, 2, 5, 3, 6, 7, 1, 8, 7, 4, 8, 5, 6, 4, 9, 7, 10, 1, 9, 8, 11, 3, 12, 11, 10, 12, 13, 1, 12, 14, 9, 10, 14, 1, 15, 6, 13, 2, 16, 3, 17, 11, 13, 5, 14, 2, 11, 6, 14, 13, 9, 15, 18, 2, 19, 5, 15, 16, 4, 17, 20, 2, 21, 3, 18, 16, 17, 5, 20, 4, 19, 6

Offset: 1

Author

Peter Kagey, Nov 11 2015

Keywords

• nonn
• look
• hear

Comments

A more detailed definition from Antti Karttunen, Dec 09 2015: (Start)

After a(1) = 1, for the next term always choose the smallest number k >= 1 such that neither k and a(n-1) nor k and a(A265400(n)) [in case A265400(n) > 0] are equal, and neither of these pairs occur anywhere adjacent to each other (horizontally or vertically) in so far constructed spiral. Here A265400(n) gives the index of the nearest horizontally or vertically adjacent inner neighbor of the n-th term in spiral, or 0 if n is one of the corner cases A033638.

The condition "... do not occur anywhere adjacent to each other (horizontally or vertically) in so far constructed spiral" can be more formally stated as: there is no such 1 < j < n, for which either the unordered pair {a(j),a(j-1)} or [in case A265400(j) > 0] also the unordered pair {a(j),a(A265400(j))} would be equal to either of the unordered pair {k,a(n-1)} or the unordered pair {k,a(A265400(n))} [in case A265400(n) > 0], where k is the term chosen for a(n). (See also my reference Scheme-implementation.)

(End)

Examples

			a(8) = 6 because pairs {1,2}, {1,4} and {1,5} already occur, the immediately adjacent terms are 1 and 3, thus neither number can be used, so the smallest usable number is 6.
a(12) = 7 because 1 and 2 are already adjacent to 8; 2, 4, 5, and 6 are already adjacent to 3.
The following illustration is the timeline of spiral's construction step-by-step:
        |      |   3  |  43  | 243  | 243  |     |  243  |  243  |  2437
    1   |  12  |  12  |  12  |  12  | 512  |     |  512  |  5128 |  5128
        |      |      |      |      |      | ... |  3671 |  3671 |  3671
        |      |      |      |      |      |     |       |       |
  a(1)=1|a(2)=2|a(3)=3|a(4)=4|a(5)=2|a(6)=5|     |a(10)=1|a(11)=8|a(12)=7
Indices of this spiral are shown below using the base-36 system, employing as its placeholder values the digits 0-9 and letter A-Z. The 1 at the center is where the spiral starts:
            ZYXWV
           HGFEDU
           I543CT
           J612BS
           K789AR
           LMNOPQ
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Antti Karttunen, R6RS-Scheme program for computing this sequence (with a naive algorithm)
• Peter Kagey, Ruby program for computing this sequence.

Crossrefs

Cf. A033638, A123663, A265400, A265579.

Cf. A272573 (analogous sequence on a hexagonal tiling).

Cf. A265414 (positions of records, where n occurs for the first time), A265415 (positions of ones).