A353590 Lexicographically earliest permutation of the nonnegative integers filling an infinite square array by falling antidiagonals so that the elements on any 2 X 2 square sum to a square.

0, 1, 2, 3, 6, 4, 5, 15, 13, 7, 8, 26, 30, 12, 9, 10, 25, 29, 45, 21, 11, 14, 38, 20, 17, 22, 23, 16, 18, 19, 61, 34, 37, 55, 31, 24, 27, 49, 51, 54, 33, 82, 35, 50, 28, 32, 75, 77, 59, 48, 44, 53, 80, 42, 36, 39, 62, 88, 69, 64, 71, 46, 57, 84, 63, 40, 41, 92, 99, 90, 97, 73, 95

Offset: 0

Author

M. F. Hasler, May 29 2022

Keywords

• nonn

Comments

In A337115 the infinite 2D lattice is filled along a square spiral satisfying the same constraint of 2 X 2 squares adding up to squares.

Examples

			The square array starts:
   0   1   3   5   8  10  14  18  27  32  ...
   2   6  15  26  25  38  19  49  75  ...
   4  13  30  29  20  61  51  77  ...
   7  12  45  17  34  54  59  ...
   9  21  22  37  33  48  ...
  11  23  55  82  44  ...
  16  31  35  53  ...
  24  50  80  ...
  28  42  ...
  36  ...
  ...
a(4) is in the second row and column. It must sum up with a(0) = 0, a(1) = 1 and a(2) = 2 to a square, the smallest possible solution is a(4) = 6.
Similarly, a(7) which is on the second row, third column, must sum up with a(1) = 1 (above to the left), a(3) = 3 (above) and a(4) = 6 (to the left) to a square; the smallest solution is a(7) = 15.
		

Crossrefs

Cf. A000290 (the squares), A337115 (same idea with square spiral instead of array by antidiagonals), A353591 (same idea with primes instead of squares).

Programs

PARI

A353590_upto(N, M=Map(), r,c, U=[-1])={vector(N, i, if(r && c, my(s=mapget(M,[r-1,c-1])+mapget(M,[r-1,c])+mapget(M,[r,c-1]), m=sqrtint(s)+1); while(setsearch(U, N=m^2-s)||N<=U[1], m+=1); U=setunion(U, [N]), N=U[1]+=1); mapput(M,[r,c], N); if(c, c--;r++, r=!c=r+1); while(#U>2 && U[2]==U[1]+1, U=U[^1]); N)}