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A353590
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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.
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1
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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)}
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CROSSREFS
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Cf. A000290 (the squares), A337115 (same idea with square spiral instead of array by antidiagonals), A353591 (same idea with primes instead of squares).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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