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%I #59 Mar 16 2024 15:22:53
%S 1,9,53,309,1801,10497,61181,356589,2078353,12113529,70602821,
%T 411503397,2398417561,13979001969,81475594253,474874563549,
%U 2767771787041,16131756158697,94022765165141,548004834832149,3194006243827753,18616032628134369,108502189524978461
%N a(n) = 6*a(n-1) - a(n-2), n >= 2, a(0)=1, a(1)=9.
%C Bisection of A048654. - Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 24 2004
%C This gives part of the (increasingly sorted) positive solutions y to the Pell equation x^2 - 2*y^2 = +7. For the x solutions see A038762. For the other part of solutions see 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.
%H Vincenzo Librandi, <a href="/A038761/b038761.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) = (9*((3+2*sqrt(2))^n -(3-2*sqrt(2))^n)-((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1)))/(4*sqrt(2)).
%F a(n) = sqrt(2*(A038762(n))^2-14)/2.
%F For n>1, a(n)-4a(n-1)=A001541(n)-A001542(n-2); e.g. 309-4*53=97=99-2. - _Charlie Marion_, Nov 12 2003
%F For n>0, a(n)=A046090(n)+A001653(n)+A001652(n-1)=A055997(n+1)+A001652(n-1); e.g., 309=120+169+20. - _Charlie Marion_, Oct 11 2006
%F G.f.: (1+3*x)/(1-6*x+x^2). - _Philippe Deléham_, Nov 03 2008
%F a(n) = third binomial transform of 1,6,8,48,64,384. - Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
%F a(n)^2 + 2^2 = A124124(2*n+1)^2 + (A124124(2*n+1)+1)^2. - _Hermann Stamm-Wilbrandt_, Aug 31 2014
%F a(n) = irrational part of z(n) = (3 + sqrt(2))*(3 + 2*sqrt(2))^n, n >= 0. z(n) gives only part of the general positive solutions to the Pell equation x^2 - 2*y^2 = 7). See the Nagell reference in A038762 on how to find z(n), and a comment above. - _Wolfdieter Lang_, Feb 05 2015
%F a(n) = S(n, 6) + 3*S(n-1, 6), n >= 0, with the Chebyshev S-polynomials evaluated at x=6. See S(n-1, 6) = A001109(n). - _Wolfdieter Lang_, Mar 30 2015
%F E.g.f.: exp(3*x)*(2*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x))/2. - _Stefano Spezia_, Mar 16 2024
%e A038762(3)^2 - 2*a(4)^2 = 2547^2 - 2*1801^2 = +7. - _Wolfdieter Lang_, Feb 05 2015
%p a[0]:=1: a[1]:=9: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..19); # _Zerinvary Lajos_, Jul 26 2006
%t LinearRecurrence[{6,-1},{1,9},40] (* _Vincenzo Librandi_, Nov 16 2011 *)
%o (Magma) I:=[1, 9]; [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) a(n)=([0,1; -1,6]^n*[1;9])[1,1] \\ _Charles R Greathouse IV_, Apr 08 2016
%Y Cf. A001109, A001541, A001542, A001652, A001653, A038762, A046090, A048654, A055997, A101386, A124124, A253811.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 02 2000
%E Edited: Replaced the unspecific Pell comment. Moved a formula from the comment section to the formula section. - _Wolfdieter Lang_, Feb 05 2015
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