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%I #90 Mar 23 2024 20:32:06
%S 1,35,1189,40391,1372105,46611179,1583407981,53789260175,
%T 1827251437969,62072759630771,2108646576008245,71631910824649559,
%U 2433376321462076761,82663163018885960315,2808114166320660573949
%N Indices of square numbers that are also hexagonal.
%C Bisection (even part) of Chebyshev sequence with Diophantine property.
%C (3*b(n))^2 - 2*(2*a(n+1))^2 = 1 with companion sequence b(n) = A077420(n), n >= 0.
%C Sequence also refers to inradius of primitive Pythagorean triangles with consecutive legs, odd followed by even. - _Lekraj Beedassy_, Apr 23 2003
%C As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> oo} a(n)/a(n-1) = (1 + sqrt(2))^4 = 17 + 12*sqrt(2). - _Ant King_, Nov 08 2011
%C Integers of the form sqrt((m+1)*(2*m+1)). The corresponding values of m form A078522. Subsequence of A284876. - _Jonathan Sondow_, Apr 07 2017
%D M. Rignaux, Query 2175, L'Intermédiaire des Mathématiciens, 24 (1917), 80.
%H Vincenzo Librandi, <a href="/A046176/b046176.txt">Table of n, a(n) for n = 1..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H E. Kilic, Y. T. Ulutas, and N. Omur, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=3.
%H Serge Perrine, <a href="http://article.scirea.org/pdf/11150.pdf">About the diophantine equation z^2 = 32y^2 - 16</a>, SCIREA Journal of Mathematics (2019) Vol. 4, Issue 5, 126-139.
%H Maciej Ulas, <a href="https://arxiv.org/abs/0811.2477">On certain diophantine equations related to triangular and tetrahedral numbers</a>, arXiv:0811.2477 [math.NT] (2008) , v_n in Theorem 5.6.
%H P. H. van der Kamp, <a href="https://doi.org/10.1007/s10208-009-9041-9">Global classification of two-component...</a>, Found. Comput. Math. 9 (5) (2009) 559-597 near Eq. (4.7).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalSquareNumber.html">Hexagonal Square Number</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (34,-1).
%F a(n) = 34*a(n-1) - a(n-2); a(0)=-1, a(1)=1.
%F a(n+1) = S(2*n, 6) = S(n, 34) + S(n-1, 34), n >= 1, with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(n, 34) = A029547(n).
%F G.f.: x*(1+x)/(1-34*x+x^2).
%F a(n+1) = Sum_{k=0..n} (-1)^k*binomial(2*n-k, k)*6^(2*(n-k)), n >= 0.
%F a(n) = A001109(2n+1). - _Lekraj Beedassy_, Apr 23 2003
%F Define f[x,s] = s*x + sqrt((s^2-1)x^2+1); f[0,s]=0. a(n) = f[f[a(n-1),3],3]. - Marcos Carreira, Dec 27 2006
%F From _Antonio Alberto Olivares_, Mar 22 2008: (Start)
%F a(n) = (sqrt(2)/8)*(3 + 2*sqrt(2))*(17 + 12*sqrt(2))^(n-1) - (sqrt(2)/8)*(3 - 2*sqrt(2))*(17 - 12*sqrt(2))^(n-1).
%F a(n) = (sqrt(2)/8)*( (17+12*sqrt(2))^(n-1/2) - (17-12*sqrt(2))^(n-1/2) ).
%F a(n) = (sqrt(2)/8)*( (3+2*sqrt(2))^(2n-1) - (3-2*sqrt(2))^(2n-1) ).
%F a(n) = (sqrt(2)/8)*( (1+sqrt(2))^(4n-2) - (1-sqrt(2))^(4n-2) ).
%F a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). (End)
%F a(n+1) = 17*a(n) + 6*sqrt(8*a(n)^2+1) for n >= 0. - _Richard Choulet_, May 01 2009
%F a(n) = b such that (-1)^(n+1) * Integral_{x=-Pi/2..Pi/2} cos((2*n-1)*x)/(3-sin(x)) dx = c + b*log(2). - _Francesco Daddi_, Aug 01 2011
%F a(n) are the nonzero integer square roots of A227970. - _Richard R. Forberg_, Aug 01 2013
%F a(n) = y/5, where y are solutions to: y^2 = 2x^2 - x - 3. - _Richard R. Forberg_, Nov 24 2013
%F a(n) = sqrt((A078522(n)+1)*(2*A078522(n)+1)). - _Jonathan Sondow_, Apr 07 2017
%F a(n) = Pell(4*n-2)/2. - _G. C. Greubel_, Jan 13 2020
%F a(n) = A001653(n)*A002315(n). - _Gerry Martens_, Mar 23 2024
%p seq( simplify(ChebyshevU(2*(n-1), 3)), n = 1..20); # _G. C. Greubel_, Jan 13 2020
%t LinearRecurrence[{34, -1}, {1, 35}, 15] (* _Ant King_, Nov 08 2011 *)
%t Fibonacci[4*Range[20] -2, 2]/2 (* _G. C. Greubel_, Jan 13 2020 *)
%o (Magma) I:=[1,35]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 22 2011
%o (PARI) vector(21, n, polchebyshev(2*(n-1), 2, 3) ) \\ _G. C. Greubel_, Jan 13 2020
%o (Sage) [lucas_number1(4*n-2, 2,-1)/2 for n in (1..20)] # _G. C. Greubel_, Jan 13 2020
%o (GAP) List([0..20], n-> Lucas(2,-1, 4*n-2)[1]/2 ); # _G. C. Greubel_, Jan 13 2020
%Y Cf. A000129, A008844, A046177, A078522, A284876.
%Y Cf. A001109, A001110 (partial sums).
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
%E Chebyshev comments from _Wolfdieter Lang_, Nov 29 2002
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