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A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2. 16

%I

%S 0,9,60,93,140,429,620,893,2576,3689,5280,15089,21576,30849,88020,

%T 125829,179876,513093,733460,1048469,2990600,4274993,6111000,17430569,

%U 24916560,35617593,101592876,145224429,207594620,592126749,846430076

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+31, y).

%C Corresponding values y of solutions (x, y) are in A157646.

%C For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436 see A118673 or A129836.

%C lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = (33 + 8*sqrt(2))/31 for n mod 3 = {1, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (1539 + 850*sqrt(2))/31^2 for n mod 3 = 0.

%D Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278. Solution appeared in Vol. 38, No. 2, May 2000, pp. 183-184.

%H G. C. Greubel, <a href="/A118674/b118674.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).

%F a(n) = 6*a(n-3) - a(n-6) + 62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.

%F G.f.: x*(9 + 51*x + 33*x^2 - 7*x^3 - 17*x^4 - 7*x^5)/((1-x)*(1 - 6*x^3 + x^6)).

%F a(3*k + 1) = 31*A001652(k) for k >= 0.

%t ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* _Jean-Fran├žois Alcover_, Dec 27 2011, after given formula *)

%t LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,9,60,93,140,429,620}, 50] (* _G. C. Greubel_, Mar 31 2018 *)

%o (PARI) {forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ",")))};

%o (MAGMA) I:=[0,9,60,93,140,429,620]; [n le 7 select I[n] else Self(n-1) - 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // _G. C. Greubel_, Mar 31 2018

%Y cf. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3 + 2*sqrt(2)), A157647 (decimal expansion of (33 + 8*sqrt(2))/31), A157648 (decimal expansion of (1539 + 850*sqrt(2))/31^2).

%K nonn

%O 1,2

%A _Mohamed Bouhamida_, May 19 2006

%E Edited by _Klaus Brockhaus_, Mar 11 2009

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Last modified May 27 21:52 EDT 2020. Contains 334671 sequences. (Running on oeis4.)