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A002350 Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is a square. A002349 gives values of y.
(Formerly M2240 N0890)
28
1, 3, 2, 1, 9, 5, 8, 3, 1, 19, 10, 7, 649, 15, 4, 1, 33, 17, 170, 9, 55, 197, 24, 5, 1, 51, 26, 127, 9801, 11, 1520, 17, 23, 35, 6, 1, 73, 37, 25, 19, 2049, 13, 3482, 199, 161, 24335, 48, 7, 1, 99, 50, 649, 66249, 485, 89, 15, 151, 19603, 530, 31, 1766319049, 63, 8, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
From A.H.M. Smeets, Nov 20 2017: (Start)
a(p*q^2) = b(p,q/gcd(A002349(p),q)) where
b(p,0) = 1, b(p,1) = a(p), b(p,i) = 2*a(p)*b(p,i-1) - b(p,i-2) for i>1. (End)
REFERENCES
A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443. (Annotated scanned copy)
L. Beeckmans, Squares expressible as sum of consecutive squares, Amer. Math. Monthly, 101 (1994), 437-442.
L. Euler, De solutione problematum diophanteorum per numeros integros (English and Latin), par. 17.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
EXAMPLE
For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).
MATHEMATICA
PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[1]], 1]; Table[ f[n], {n, 0, 65}]
Table[If[! IntegerQ[Sqrt[k]], {k, FindInstance[x^2 - k*y^2 == 1 && x > 0 && y > 0, {x, y}, Integers]}, Nothing], {k, 2, 80}][[All, 2, 1, 1, 2]] (* Horst H. Manninger, Mar 23 2021 *)
CROSSREFS
Sequence in context: A193791 A160760 A152860 * A109267 A185416 A193918
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified April 16 10:08 EDT 2024. Contains 371698 sequences. (Running on oeis4.)