A257781 Positive integers whose square is the sum of 50 consecutive squares.

245, 385, 495, 655, 795, 1055, 1365, 2205, 2855, 3795, 4615, 6135, 7945, 12845, 16635, 22115, 26895, 35755, 46305, 74865, 96955, 128895, 156755, 208395, 269885, 436345, 565095, 751255, 913635, 1214615, 1573005, 2543205, 3293615, 4378635, 5325055, 7079295

Offset: 1

Author

Colin Barker, May 08 2015

Keywords

• nonn
• easy

Comments

Positive integers x in the solutions to 2*x^2-100*y^2-4900*y-80850 = 0.

Examples

			245 is in the sequence because 245^2 = 60025 = 7^2+8^2+...+56^2.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,6,0,0,0,0,0,-1).

Crossrefs

Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257823-A257828.

Programs

Magma

I:=[245,385,495,655,795,1055,1365,2205,2855,3795, 4615,6135]; [n le 12 select I[n] else 6*Self(n-6)-Self(n-12): n in [1..40]]; // Vincenzo Librandi, May 11 2015

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -1}, {245, 385, 495, 655, 795, 1055, 1365, 2205, 2855, 3795, 4615, 6135}, 50] (* Vincenzo Librandi, May 11 2015 *)
Select[Sqrt[Total/@Partition[Range[10^6]^2,50,1]],IntegerQ] (* Harvey P. Dale, Aug 07 2025 *)

PARI

Vec(-5*x*(39*x^11 +31*x^10 +27*x^9 +23*x^8 +21*x^7 +21*x^6 -211*x^5 -159*x^4 -131*x^3 -99*x^2 -77*x -49) / ((x^6 -2*x^3 -1)*(x^6 +2*x^3 -1)) + O(x^100))

Formula

a(n) = 6*a(n-6)-a(n-12).

G.f.: -5*x*(39*x^11 +31*x^10 +27*x^9 +23*x^8 +21*x^7 +21*x^6 -211*x^5 -159*x^4 -131*x^3 -99*x^2 -77*x -49) / ((x^6 -2*x^3 -1)*(x^6 +2*x^3 -1)).