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 A272259 Irregular triangle read by rows: Row n>=32 gives the smallest square loop, i.e., lexicographically earliest circular permutation of length n such that any two adjacent numbers sum to a perfect square. 2
 1, 8, 28, 21, 4, 32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 31, 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15, 1, 8, 28, 21, 4, 32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 20, 29, 7, 18, 31, 33, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15, 1, 3, 13, 12, 4, 32, 17, 8, 28, 21, 15, 34, 30, 19, 6, 10, 26, 23, 2, 14, 22, 27, 9, 16, 33, 31, 18, 7, 29, 20, 5, 11, 25, 24 (list; graph; refs; listen; history; text; internal format)
 OFFSET 32,2 COMMENTS T(n) gives the smallest Hamiltonian cycle in the corresponding undirected unweighted graph with n vertices and edges satisfying the square sum condition, so this is also a solution to the Traveling Salesman Problem. There are no circular solutions for n < 32. T(32) = A112663(k-1), 1 <= k <= 32. Row n has length n, and we start with row n = 32. LINKS Martin Renner, Table of n, a(n) for n = 32..663 EXAMPLE Table starts with n = 32: 1, 8, 28, 21, 4, 32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 31, 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15. n = 33: 1, 8, 28, 21, 4, 32, 17, 19, 30, 6, 3, 13, 12, 24, 25, 11, 5, 20, 29, 7, 18, 31, 33, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15. MAPLE with(GraphTheory): n:=32; # Vertices from 1 to n E:={}: # Edges for a from 1 to n do   for b from a+1 to n do     if type(sqrt(a+b), integer) then E:={op(E), {a, b}}: fi:   od: od: G:=Graph(E); T||n:=TravelingSalesman(G)[2, 1..n]; CROSSREFS Cf. A071984, A112663. Sequence in context: A088029 A126815 A053619 * A112663 A220288 A201101 Adjacent sequences:  A272256 A272257 A272258 * A272260 A272261 A272262 KEYWORD nonn,tabf AUTHOR Martin Renner, Apr 23 2016 STATUS approved

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Last modified May 23 20:29 EDT 2022. Contains 353993 sequences. (Running on oeis4.)