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 A160206 Positive numbers y such that y^2 is of the form x^2+(x+857)^2 with integer x. 4
 697, 857, 1117, 3065, 4285, 6005, 17693, 24853, 34913, 103093, 144833, 203473, 600865, 844145, 1185925, 3502097, 4920037, 6912077, 20411717, 28676077, 40286537, 118968205, 167136425, 234807145, 693397513, 974142473, 1368556333 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (-185, a(1)) and (A129857(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2). lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {0, 2}. lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 1. LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 FORMULA a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=697, a(2)=857, a(3)=1117, a(4)=3065, a(5)=4285, a(6)=6005. G.f.: (1-x)*(697+1554*x+2671*x^2+1554*x^3+697*x^4)/(1-6*x^3+x^6). a(3*k-1) = 857*A001653(k) for k >= 1. EXAMPLE (-185, a(1)) = (-185, 697) is a solution: (-185)^2+(-185+857)^2 = 34225+451584 = 485809 = 697^2. (A129857(1), a(2)) = (0, 857) is a solution: 0^2+(0+857)^2 = 734449 = 857^2. (A129857(3), a(4)) = (1696, 3065) is a solution: 1696^2+(1696+857)^2 = 2876416+6517809 = 9394225 = 3065^2. MATHEMATICA LinearRecurrence[{0, 0, 6, 0, 0, -1}, {697, 857, 1117, 3065, 4285, 6005}, 50] (* G. C. Greubel, May 14 2018 *) PROG (PARI) {forstep(n=-188, 10000000, [3, 1], if(issquare(2*n^2 +1714*n +734449, &k), print1(k, ", ")))} (PARI) x='x+O('x^30); Vec((1-x)*(697+1554*x+2671*x^2+1554*x^3 +697*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, May 14 2018 (MAGMA) I:=[697, 857, 1117, 3065, 4285, 6005]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..40]]; // G. C. Greubel, May 14 2018 CROSSREFS Cf. A129857, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2). Sequence in context: A282549 A069330 A218156 * A111105 A137559 A185377 Adjacent sequences:  A160203 A160204 A160205 * A160207 A160208 A160209 KEYWORD nonn AUTHOR Klaus Brockhaus, May 18 2009 STATUS approved

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Last modified August 18 08:39 EDT 2018. Contains 313823 sequences. (Running on oeis4.)