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 A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)*b(n) - G(n)*b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n). 4
 1, 2, 5, 3, 3, 27, 7, 37, 4, 4, 171, 22, 9, 14, 1193, 5, 5, 553, 16, 6173, 11, 45, 143, 849, 6, 6, 18339, 94, 1893, 103, 13, 33, 2353, 115, 12703, 7, 7, 67115, 701, 73, 59, 1891117, 15, 551427, 23, 49771, 39, 4105015, 8, 8, 24673, 41, 75585293, 25, 9095891, 989, 17, 386, 6445, 87, 771, 1385 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This equation can also be written as (2*a(n) - b(n))^2 - D(n)*b(n)^2 = +4 or -4 with D(n) := A077425(n) = 1 + 4*G(n). This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations. For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357. REFERENCES O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108). LINKS Table of n, a(n) for n=1..62. Wolfdieter Lang, Pairs of smallest positive solutions for +1 and -1 if they exist. FORMULA a(n) = (A078361(n) + A077058(n)) / 2. [Max Alekseyev, Feb 06 2010] MATHEMATICA g[n_] := Ceiling[Sqrt[n]] + n - 1; r[n_] := Reduce[an > 0 && bn > 0 && (an ^2 - an*bn - g[n]*bn^2 == 1 || an^2 - an*bn - g[n]*bn^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}], 1], an | bn]; a[n_] := a[n] = Min[ab[n][[All, 1]]]; Table[Print[{n, a[n]}]; a[n], {n, 1, 62}] (* Jean-François Alcover, Oct 04 2012 *) CROSSREFS Cf. A077427, A217470. Sequence in context: A281300 A115320 A073480 * A359684 A030660 A253720 Adjacent sequences: A077054 A077055 A077056 * A077058 A077059 A077060 KEYWORD nonn AUTHOR Wolfdieter Lang, Nov 29 2002 EXTENSIONS More terms from Max Alekseyev, Feb 06 2010 a(9), a(33), a(54) corrected (after notice by Jean-François Alcover); a(58) through a(62) added. - Wolfdieter Lang, Oct 04 2012 STATUS approved

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Last modified February 26 19:18 EST 2024. Contains 370352 sequences. (Running on oeis4.)