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 A160583 Positive numbers y such that y^2 is of the form x^2+(x+521)^2 with integer x. 3
 445, 521, 629, 2041, 2605, 3329, 11801, 15109, 19345, 68765, 88049, 112741, 400789, 513185, 657101, 2335969, 2991061, 3829865, 13615025, 17433181, 22322089, 79354181, 101608025, 130102669, 462510061, 592214969, 758293925, 2695706185, 3451681789, 4419660881 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (-84, a(1)) and (A129725(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2). lim_{n -> infinity} a(n)/a(n-1) = (537+92*sqrt(2))/521 for n mod 3 = {0, 2}. lim_{n -> infinity} a(n)/a(n-1) = (520659+314170*sqrt(2))/521^2 for n mod 3 = 1. LINKS Table of n, a(n) for n=1..30. Index entries for linear recurrences with constant coefficients, signature (0, 0, 6, 0, 0, -1). FORMULA a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=445, a(2)=521, a(3)=629, a(4)=2041, a(5)=2605, a(6)=3329. G.f.: (1-x)*(445+966*x+1595*x^2+966*x^3+445*x^4) / (1-6*x^3+x^6). a(3*k-1) = 521*A001653(k) for k >= 1. EXAMPLE (-84, a(1)) = (-84, 445) is a solution: (-84)^2+(-84+521)^2 = 7056+190969 = 198025 = 445^2. (A129725(1), a(2)) = (0, 521) is a solution: 0^2+(0+521)^2 = 271441 = 521^2. (A129725(3), a(4)) = (1159, 2041) is a solution: 1159^2+(1159+521)^2 = 1343281+2822400 = 4165681 = 2041^2. MATHEMATICA LinearRecurrence[{0, 0, 6, 0, 0, -1}, {445, 521, 629, 2041, 2605, 3329}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *) PROG (PARI) {forstep(n=-84, 10000000, [3, 1], if(issquare(2*n^2+1042*n+271441, &k), print1(k, ", ")))} CROSSREFS Cf. A129725, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2). Sequence in context: A028460 A292139 A105985 * A340151 A025354 A025346 Adjacent sequences: A160580 A160581 A160582 * A160584 A160585 A160586 KEYWORD nonn AUTHOR Klaus Brockhaus, Jun 08 2009 STATUS approved

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Last modified September 22 11:05 EDT 2023. Contains 365520 sequences. (Running on oeis4.)