A308776 Counterclockwise square spiral of distinct positive integers constructed by greedy algorithm, such that for k = 1..5, every term equal to k mod 5 (except k itself) is vertically or horizontally adjacent to a cell already holding a value equal to k mod 5.

1, 2, 3, 4, 5, 6, 11, 16, 7, 12, 17, 8, 13, 18, 9, 10, 15, 20, 21, 26, 31, 36, 41, 22, 27, 32, 37, 42, 23, 28, 33, 38, 43, 14, 19, 24, 29, 25, 30, 35, 40, 45, 50, 46, 51, 56, 47, 52, 57, 62, 67, 72, 77, 48, 53, 58, 63, 68, 73, 78, 34, 39, 44, 49, 54, 59, 55

Offset: 1

Author

Rémy Sigrist, Jun 24 2019

Keywords

• nonn

Comments

For k = 1..5, let M_k be the set of lattice points with a value equal to k mod 5; the sets M_1, ..., M_5 form 5 arms that spiral around the origin (see representation in Links section). As a consequence, the sequence alternates runs of terms in arithmetic progression with first difference 5.

This sequence appears to be a permutation of the natural numbers.

This sequence has similarities with A308505.

Examples

			The spiral begins:
   89---84---79---74---69---64--123--118--113--108--103
    |                                                 |
   94   54---49---44---39---34---78---73---68---63   98
    |    |                                       |    |
   99   59   29---24---19---14---43---38---33   58   93
    |    |    |                             |    |    |
   95   55   25   15---10----9---18---13   28   53   88
    |    |    |    |                   |    |    |    |
  100   60   30   20    5----4----3    8   23   48   83
    |    |    |    |    |         |    |    |    |    |
  105   65   35   21    6    1----2   17   42   77  112
    |    |    |    |    |              |    |    |    |
  110   70   40   26   11---16----7---12   37   72  107
    |    |    |    |                        |    |    |
  115   75   45   31---36---41---22---27---32   67  102
    |    |    |                                  |    |
  120   80   50---46---51---56---47---52---57---62   97
    |    |                                            |
  125   85---90---61---66---71---76---81---82---87---92
    |
  130--135--140---86---91---96--101--106--111--116--117
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10201
• Rémy Sigrist, Colored representation of the spiral for -500 <= x <= 500 and -500 <= y <= 500 (where the color is function of a(n) mod 5)
• Rémy Sigrist, Colored scatterplot of (n, a(n)-n) for n = 1..40401 (where the color is function of a(n) mod 5)
• Rémy Sigrist, PARI program for A308776

Crossrefs

Cf. A010874, A308505.

Programs

PARI

See Links section.