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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
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
Sequence in context: A281300 A115320 A073480 * A359684 A030660 A253720
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.)