Numbers beginning 50 consecutive numbers whose squares sum to a square.
7, 28, 44, 67, 87, 124, 168, 287, 379, 512, 628, 843, 1099, 1792, 2328, 3103, 3779, 5032, 6524, 10563, 13687, 18204, 22144, 29447, 38143, 61684, 79892, 106219, 129183, 171748, 222432, 359639, 465763, 619208, 753052, 1001139, 1296547, 2096248, 2714784
1
Numbers satisfy a 13-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 7^2 + 8^2 + … + 56^2 = 245^2.
(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[#, # + 50 - 1]]] &]
(Mma) LinearRecurrence[{1, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, -1, 1}, {7, 28, 44, 67, 87, 124, 168, 287, 379, 512, 628, 843, 1099}, 50]
Cf. A001032, A001652, A106521, A094196, S000277-S000283, S000284.
nonn
T. D. Noe, Oct 08 2014