Numbers beginning 338 consecutive numbers whose squares sum to a square.
27, 52, 175, 216, 1036, 1175, 1872, 2107, 6863, 7672, 11731, 13100, 40816, 45531, 69188, 77167, 238707, 266188, 404071, 450576, 1392100, 1552271, 2355912, 2626963, 8114567, 9048112, 13732075, 15311876, 47295976, 52737075, 80037212, 89244967, 275661963
1
Numbers satisfy a 9-th order linear recurrence.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Moshe Laub, Squares Expressible as a Sum of n Consecutive Squares, Advanced Problem 6552, Amer. Math. Monthly 97 (1990), 622-625.
The smallest example is 27^2 + 28^2 + … + 364^2 = 4017^2.
(Mma) g[m0_, m1_] := (1 - m0 + m1) (-m0 + 2 m0^2 + m1 + 2 m0 m1 + 2 m1^2)/6; Select[Range[100000], IntegerQ[Sqrt[g[#, # + 338 - 1]]] &]
(Mma) LinearRecurrence[{1, 0, 0, 6, -6, 0, 0, -1, 1}, {27, 52, 175, 216, 1036, 1175, 1872, 2107, 6863}, 50]
Cf. A001032, A001652, A106521, A094196, S000277-S000283, S000284.
nonn
T. D. Noe, Oct 10 2014