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Search: id:A033314
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| A033314 |
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Least D in the Pellian x^2 - D*y^2 = 1 for which x has least solution n. |
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6
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| 3, 2, 15, 6, 35, 12, 7, 5, 11, 30, 143, 42, 195, 14, 255, 18, 323, 10, 399, 110, 483, 33, 23, 39, 27, 182, 87, 210, 899, 60, 1023, 17, 1155, 34, 1295, 38, 1443, 95, 1599, 105, 1763, 462, 215, 506, 235, 138, 47, 96, 51, 26, 2703, 78, 2915, 21, 3135, 203, 3363
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OFFSET
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2,1
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COMMENT
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The i-th solution pair V(i) = [x(i), y(i)] to the Pellian x^2 - D*y^2 = 1 for a given least solution x(1) = n may be generated through the recurrence V(i+2) = 2*n*V(i+1) - V(i) taking V(0) = [1, 0] and V(1) = [n, sqrt{(n^2-1)/a(n)}]. V(i) stands for the numerator and denominator of the 2i-th convergent of the continued fraction expansion of sqrt(D).
Thus setting n = 3, for instance, we have D = a(3) = 2 and V(1) = [3, 2] so that along with V(0) = [1, 0] recurrence V(i+2) = 6*V(i+1)-V(i) generates [A001333(2k), A000129(2k)]. Similarly, setting n = 9 generates [A023039, A060645], respectively the numerator and denominator of the 2i-th convergent of sqrt{a(9)}, i.e. sqrt{5}. - Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 26 2002
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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CROSSREFS
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Cf. A033313.
Sequence in context: A055234 A086485 A068310 this_sequence A070260 A142705 A072346
Adjacent sequences: A033311 A033312 A033313 this_sequence A033315 A033316 A033317
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KEYWORD
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nonn
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com)
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