cp's OEIS Frontend

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.

%I A353590 #15 Jun 14 2022 06:56:56
%S A353590 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,
%T A353590 23,16,18,19,61,34,37,55,31,24,27,49,51,54,33,82,35,50,28,32,75,77,59,
%U A353590 48,44,53,80,42,36,39,62,88,69,64,71,46,57,84,63,40,41,92,99,90,97,73,95
%N 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.
%C A353590 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.
%e A353590 The square array starts:
%e A353590    0   1   3   5   8  10  14  18  27  32  ...
%e A353590    2   6  15  26  25  38  19  49  75  ...
%e A353590    4  13  30  29  20  61  51  77  ...
%e A353590    7  12  45  17  34  54  59  ...
%e A353590    9  21  22  37  33  48  ...
%e A353590   11  23  55  82  44  ...
%e A353590   16  31  35  53  ...
%e A353590   24  50  80  ...
%e A353590   28  42  ...
%e A353590   36  ...
%e A353590   ...
%e A353590 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.
%e A353590 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.
%o A353590 (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)}
%Y A353590 Cf. A000290 (the squares), A337115 (same idea with square spiral instead of array by antidiagonals), A353591 (same idea with primes instead of squares).
%K A353590 nonn
%O A353590 0,3
%A A353590 _M. F. Hasler_, May 29 2022