Numbers beginning 33 consecutive numbers whose squares sum to a square.
7, 27, 60, 181, 227, 612, 1085, 1985, 3492, 9047, 11161, 28860, 50607, 91987, 161276, 416685, 513883, 1327652, 2327541, 4230121, 7415908, 19159167, 23628161, 61043836, 107016983, 194494283, 340971196, 880905701, 1086382227, 2806689508, 4920454381
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 + … + 39^2 = 143^2.
(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[#, # + 33 - 1]]] &]
(Mma) LinearRecurrence[{1, 0, 0, 0, 0, 46, -46, 0, 0, 0, 0, -1, 1}, {7, 27, 60, 181, 227, 612, 1085, 1985, 3492, 9047, 11161, 28860, 50607}, 50]
Cf. A001032, A001652, A106521, A094196, S000277-S000283, S000284.
nonn
T. D. Noe, Oct 08 2014