A075843 Numbers k such that 99*k^2 + 1 is a square.

0, 1, 20, 399, 7960, 158801, 3168060, 63202399, 1260879920, 25154396001, 501827040100, 10011386405999, 199725901079880, 3984506635191601, 79490406802752140, 1585823629419851199, 31636982181594271840

Offset

0,3

Comments

From Wolfdieter Lang, Nov 08 2002: (Start)

Chebyshev's polynomials U(n,x) evaluated at x=10.

The a(n) give all (unsigned, integer) solutions of Pell equation b(n)^2 - 99*a(n)^2 = +1 with b(n)= A001085(n). (End)

For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 20's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imagianry unit). - John M. Campbell, Jul 08 2011

For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,19}. - Milan Janjic, Jan 25 2015

References

A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.

L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.

Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Tanya Khovanova, Recursive Sequences

J. J. O'Connor and E. F. Robertson, Pell's Equation

Eric Weisstein's World of Mathematics, Pell Equation.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (20,-1).

Formula

a(n) = ((10+3*sqrt(11))^n - (10-3*sqrt(11))^n) / (6*sqrt(11)).

a(n) = 20*a(n-1) - a(n-2), n>=1, a(0)=0, a(1)=1.

a(n) = S(n-1, 20), with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. S(-1, x) := 0. See A049310.

G.f.: x/(1 - 20*x + x^2).

a(n) = sqrt((A001085(n)^2 - 1)/99).

Lim_{n->inf.} a(n)/a(n-1) = 10 + 3*sqrt(11).

a(n+1) = Sum_{k=0..n} A101950(n,k)*19^k. - Philippe Deléham, Feb 10 2012

Product_{n>=1} (1 + 1/a(n)) = 1/3*(3 + sqrt(11)). - Peter Bala, Dec 23 2012

Product_{n>=2} (1 - 1/a(n)) = 3/20*(3 + sqrt(11)). - Peter Bala, Dec 23 2012

Maple

seq( simplify(ChebyshevU(n-1, 10)), n=0..20); # G. C. Greubel, Dec 22 2019

Mathematica

Table[GegenbauerC[n-1, 1, 10], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[x/(1-20x+x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
ChebyshevU[Range[22] -2, 10] (* G. C. Greubel, Dec 22 2019 *)
LinearRecurrence[{20, -1}, {0, 1}, 20] (* Harvey P. Dale, Dec 03 2023 *)

Prog

(Sage) [lucas_number1(n, 20, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 25 2008
(Sage) [chebyshev_U(n-1, 10) for n in (0..20)] # G. C. Greubel, Dec 22 2019
(Magma) I:=[0, 1]; [n le 2 select I[n] else 20*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012
(PARI) vector( 22, n, polchebyshev(n-2, 2, 10) ) \\ G. C. Greubel, Dec 22 2019
(GAP) m:=10;; a:=[0, 1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

Crossrefs

Cf. A001084.

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), this sequence (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).

Cf. A323182.

Sequence in context: A358853 A158534 A171325 * A208072 A208122 A207372

Adjacent sequences: A075840 A075841 A075842 * A075844 A075845 A075846

Keyword

nonn,easy

Author

Gregory V. Richardson, Oct 14 2002

Status

approved