A354375 Square spiral on a 2D square lattice, one term per cell, starting at the origin with 0; the digits of the four integers forming any 2 X 2 square add up to a square and those sums themselves form another infinite 2D square lattice with the same property.

0, 1, 2, 6, 3, 999, 4, 5, 12, 7, 799, 8, 9, 89, 29, 79, 10, 88, 8999, 69, 11, 78, 39, 97, 19, 13, 87, 7999, 59, 14, 15, 169, 39999, 68, 49999, 699, 16, 22, 96, 159, 178, 21, 17, 599, 59999, 49, 58999, 168, 25, 18, 187, 100, 4999, 20, 177, 28, 23, 186, 89999, 99999, 199999, 98999, 9999, 77, 24, 27

Offset

1,3

Comments

This is the earliest permutation of the nonnegative integers with this property.

Links

Table of n, a(n) for n=1..66.

Example

The spiral begins:
.
     11--78--39--97--19--13
      |                   |
     69   4---5--12---7  87
      |   |           |   |
   8999  999  0---1  799 7999
      |   |       |   |   |
     88   3---6---2   8  59
      |               |   |
     10--79--29--89---9  14
                          |
           ... 39999-169-15
.
The digits of the four integers inside each of the four 2 X 2 squares that contain the initial 0 add up to a square: 0 + 1 + 2 + 6 = 9, 0 + 6 + 3 + (9+9+9) = 36, 0 + 999 + 4 + 5 = 36, 0 + 5 + (1+2) + 1 = 9. This is true for any 2 X 2 square on the (infinite) grid; the digits of the upper right corner add up to 36, for instance: (1+9) + (1+3) + (8+7) + 7 = 36; the lower right 2 X 2 square produces 36 = 9 + (1+4) + (1+5) + (1+6+9); etc.
All those successive "square sums" form the hereunder "second-level" spiral:
.
       36---9--36--81
        |           |
       36   9--36  81
        |       |   |
       36--36--36  36
                    |
           ... 81--36
.
Though the terms of this new spiral are not distinct (only multiples of 9), the sum of the digits inside any 2 X 2 square is a square again; the upper left 2 X 2 square produces for instance the square 36 = (3+6) + 9 + 9 + (3+6); the lower left 2 X 2 square produces the square 36 again = (3+6) + 9 + (3+6) + (3+6); the lower right 2 X 2 square produces also the square 36 = (3+6) + (3+6) + (3+6) + (8+1); the initial "center square" produces the same 36 = 9 + (3+6) + (3+6) + (3+6); etc.

Crossrefs

Cf. A337115, A337116, A337117, A337368, A354372, A354373, A354374.

Sequence in context: A101819 A177761 A128192 * A077750 A332253 A247493

Adjacent sequences: A354372 A354373 A354374 * A354376 A354377 A354378

Keyword

nonn,base

Author

Eric Angelini and Carole Dubois, May 24 2022

Status

approved