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The first of 50 consecutive positive integers the sum of the squares of which is a square.
4

%I #19 Oct 22 2016 10:37:33

%S 7,28,44,67,87,124,168,287,379,512,628,843,1099,1792,2328,3103,3779,

%T 5032,6524,10563,13687,18204,22144,29447,38143,61684,79892,106219,

%U 129183,171748,222432,359639,465763,619208,753052,1001139,1296547,2096248,2714784

%N The first of 50 consecutive positive integers the sum of the squares of which is a square.

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

%C Numbers n such that 40425 + 2450*n + 50*n^2 is a square. - _Harvey P. Dale_, Oct 22 2016

%H Colin Barker, <a href="/A269451/b269451.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,6,-6,0,0,0,0,-1,1).

%F G.f.: x*(7+21*x+16*x^2+23*x^3+20*x^4+37*x^5+2*x^6-7*x^7-4*x^8-5*x^9-4*x^10-7*x^11-x^12) / ((1-x)*(1+2*x^3-x^6)*(1-2*x^3-x^6)).

%e 7 is in the sequence because sum(k=7, 56, k^2) = 60025 = 245^2.

%t Select[Range[3*10^6],IntegerQ[Sqrt[40425+2450#+50#^2]]&] (* or *) 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},40] (* _Harvey P. Dale_, Oct 22 2016 *)

%o (PARI) Vec(x*(7+21*x+16*x^2+23*x^3+20*x^4+37*x^5+2*x^6-7*x^7-4*x^8-5*x^9-4*x^10-7*x^11-x^12) / ((1-x)*(1+2*x^3-x^6)*(1-2*x^3-x^6)) + O(x^40))

%Y Cf. A001032, A001652, A094196, A106521, A257781, A269447, A269448, A269449.

%K nonn,easy,less

%O 1,1

%A _Colin Barker_, Feb 27 2016