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%I #55 Mar 16 2024 15:22:00
%S 3,13,75,437,2547,14845,86523,504293,2939235,17131117,99847467,
%T 581953685,3391874643,19769294173,115223890395,671574048197,
%U 3914220398787,22813748344525,132968269668363,774995869665653,4517006948325555,26327045820287677,153445267973400507
%N a(n) = 6*a(n-1) - a(n-2) for n >= 2, with a(0)=3, a(1)=13.
%C This gives part of the (increasingly sorted) positive solutions x to the Pell equation x^2 - 2*y^2 = +7. For the y solutions see A038761. The other part of solutions is found in A101386 and A253811. - _Wolfdieter Lang_, Feb 05 2015
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
%H Vincenzo Librandi, <a href="/A038762/b038762.txt">Table of n, a(n) for n = 0..400</a>
%H M. J. DeLeon, <a href="http://www.fq.math.ca/Scanned/14-5/deleon.pdf">Pell's Equation and Pell Number Triples</a>, Fib. Quart., 14(1976), pp. 456-460.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = sqrt(2*(A038761(n))^2+7).
%F a(n) = (13*((3+2*sqrt(2))^n -(3-2*sqrt(2))^n)-3*((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1)))/(4*sqrt(2)).
%F a(n) = A077443(2n) = A038725(n)+A038725(n+1).
%F a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3); a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^(n-1)+(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^(n-1). - _Antonio Alberto Olivares_, Apr 20 2008
%F G.f.: (3-5*x)/(1-6*x+x^2). - _Philippe Deléham_, Nov 03 2008, corrected by _R. J. Mathar_, Nov 06 2011
%F a(n) = -5*A001109(n) +3*A001109(n+1). - _R. J. Mathar_, Nov 06 2011
%F a(n) = rational part of z(n) = (3 + sqrt(2))*(3 + 2*sqrt(2))^n, n >= 0. z(n) gives only one part of the positive solutions to the Pell equation x^2 - 2*y^2 = 7. See the Nagell reference on how to find z(n), and a comment above. - _Wolfdieter Lang_, Feb 05 2015
%F E.g.f.: exp(3*x)*(3*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)). - _Stefano Spezia_, Mar 16 2024
%e a(3)^2 - 2*A038761(3)^2 = 437^2 - 2*309^2 = +7. - _Wolfdieter Lang_, Feb 05 2015
%t LinearRecurrence[{6,-1},{3,13},40] (* _Vincenzo Librandi_, Nov 16 2011 *)
%o (Magma) I:=[3, 13]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Nov 16 2011
%o (PARI) x='x+O('x^30); Vec((3-5*x)/(1-6*x+x^2)) \\ _G. C. Greubel_, Jul 26 2018
%Y Cf. A001109, A038725, A038761, A077443, A101386, A253811.
%K easy,nonn
%O 0,1
%A _Barry E. Williams_, May 03 2000
%E More terms from _James A. Sellers_, May 04 2000
%E Unspecific Pell comment replaced by _Wolfdieter Lang_, Feb 05 2015
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