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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.

%I A354375 #9 Jun 25 2022 22:04:28
%S A354375 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,
%T A354375 13,87,7999,59,14,15,169,39999,68,49999,699,16,22,96,159,178,21,17,
%U A354375 599,59999,49,58999,168,25,18,187,100,4999,20,177,28,23,186,89999,99999,199999,98999,9999,77,24,27
%N 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.
%C A354375 This is the earliest permutation of the nonnegative integers with this property.
%e A354375 The spiral begins:
%e A354375 .
%e A354375      11--78--39--97--19--13
%e A354375       |                   |
%e A354375      69   4---5--12---7  87
%e A354375       |   |           |   |
%e A354375    8999  999  0---1  799 7999
%e A354375       |   |       |   |   |
%e A354375      88   3---6---2   8  59
%e A354375       |               |   |
%e A354375      10--79--29--89---9  14
%e A354375                           |
%e A354375            ... 39999-169-15
%e A354375 .
%e A354375 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.
%e A354375 All those successive "square sums" form the hereunder "second-level" spiral:
%e A354375 .
%e A354375        36---9--36--81
%e A354375         |           |
%e A354375        36   9--36  81
%e A354375         |       |   |
%e A354375        36--36--36  36
%e A354375                     |
%e A354375            ... 81--36
%e A354375 .
%e A354375 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.
%Y A354375 Cf.  A337115, A337116, A337117, A337368, A354372, A354373, A354374.
%K A354375 nonn,base
%O A354375 1,3
%A A354375 _Eric Angelini_ and _Carole Dubois_, May 24 2022