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A097315 Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n) = A097314(n), n >= 0. 21
1, 37, 1405, 53353, 2026009, 76934989, 2921503573, 110940200785, 4212806126257, 159975692596981, 6074863512559021, 230684837784645817, 8759948972303982025, 332647376109766671133, 12631840343198829521029, 479677285665445755127969, 18215105014943739865341793, 691694313282196669127860165 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Hypotenuses of primitive Pythagorean triples in A195616 and A195617. - Clark Kimberling, Sep 22 2011
LINKS
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
FORMULA
a(n) = S(n, 38) - S(n-1, 38) = T(2*n+1, sqrt(10))/sqrt(10), with Chebyshev polynomials of the second and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 6*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-38*x+x^2).
a(n) = 38*a(n-1) - a(n-2) for n > 1. - Philippe Deléham, Nov 18 2008
a(n) = sqrt(2+(19-6*sqrt(10))^(1+2*n) + (19+6*sqrt(10))^(1+2*n))/(2*sqrt(10)). - Gerry Martens, Jun 04 2015
a(n) = A078987(n) - A078987(n-1). - R. J. Mathar, Dec 05 2015
a(n) = A005668(2*n+1). - Michael Somos, Feb 24 2023
EXAMPLE
(x,y) = (3,1), (117,37), (4443,1405), ... give the positive integer solutions to x^2 - 10*y^2 = -1.
G.f. = 1 + 37*x + 1405*x^2 + 53353*x^3 + ... - Michael Somos, Feb 24 2023
MATHEMATICA
CoefficientList[Series[(1-x)/(1-38x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Feb 04 2017 *)
LinearRecurrence[{38, -1}, {1, 37}, 21] (* G. C. Greubel, Aug 01 2019 *)
PROG
(PARI) Vec((1-x)/(1-38*x+x^2) + O(x^20)) \\ Michel Marcus, Jun 04 2015
(Magma) I:=[1, 37]; [n le 2 select I[n] else 38*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-38*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 37];; for n in [3..20] do a[n]:=38*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
CROSSREFS
Row 3 of array A188647.
Cf. A221874.
Cf. similar sequences listed in A238379.
Sequence in context: A189061 A009981 A370334 * A158741 A094490 A231543
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 31 2004
EXTENSIONS
Typo in recurrence formula corrected by Laurent Bonaventure (bonave(AT)free.fr), Oct 03 2010
More terms added by Indranil Ghosh, Feb 04 2017
STATUS
approved

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Last modified April 13 18:46 EDT 2024. Contains 371644 sequences. (Running on oeis4.)