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 A046176 Indices of square numbers that are also hexagonal. 19
 1, 35, 1189, 40391, 1372105, 46611179, 1583407981, 53789260175, 1827251437969, 62072759630771, 2108646576008245, 71631910824649559, 2433376321462076761, 82663163018885960315, 2808114166320660573949 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Bisection (even part) of Chebyshev sequence with Diophantine property. (3*b(n))^2 - 2*(2*a(n+1))^2 = 1 with companion sequence b(n)=A077420(n), n>=0. Sequence also refers to inradius of primitive Pythagorean triangles with consecutive legs, odd followed by even. - Lekraj Beedassy, Apr 23 2003 As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (1 + sqrt(2))^4 = 17 + 12 * sqrt(2). - Ant King, Nov 08 2011 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..200 Tanya Khovanova, Recursive Sequences E. Kilic, Y. T. Ulutas, N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=3. Serge Perrine, About the diophantine equation z^2 = 32y^2 - 16, SCIREA Journal of Mathematics (2019) Vol. 4, Issue 5, 126-139. P. H. van der Kamp, Global classification of two-component..., Found. Comput. Math. 9 (5) (2009) 559-597 near Eq. (4.7) Eric Weisstein's World of Mathematics, Hexagonal Square Number. Index entries for linear recurrences with constant coefficients, signature (34,-1). FORMULA a(n) = 34*a(n-1) - a(n-2), a(0)=-1, a(1)=1. 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). G.f.: x*(1+x)/(1-34*x+x^2). a(n+1) = Sum_{k=0..n} (-1)^k*binomial(2*n-k, k)*6^(2*(n-k)), n>=0. a(n) = A001109(2n+1). - Lekraj Beedassy, Apr 23 2003 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 From Antonio Alberto Olivares, Mar 22 2008: (Start) 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). a(n) = (sqrt(2)/8)*( (17+12*sqrt(2))^(n-1/2) - (17-12*sqrt(2))^(n-1/2) ). a(n) = (sqrt(2)/8)*( (3+2*sqrt(2))^(2n-1) - (3-2*sqrt(2))^(2n-1) ). a(n) = (sqrt(2)/8)*( (1+sqrt(2))^(4n-2) - (1-sqrt(2))^(4n-2) ). a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). (End) a(n+1) = 17*a(n) + 6*sqrt(8*a(n)^2+1) for n>=0. - Richard Choulet, May 01 2009 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 a(n) are the nonzero integer square roots of A227970. - Richard R. Forberg, Aug 01 2013 a(n) = y/5, where y are solutions to: y^2 = 2x^2 - x - 3. - Richard R. Forberg, Nov 24 2013 a(n) = sqrt((A078522(n)+1)*(2*A078522(n)+1)). - Jonathan Sondow, Apr 07 2017 a(n) = Pell(4*n-2)/2. - G. C. Greubel, Jan 13 2020 MAPLE seq( simplify(ChebyshevU(2*(n-1), 3)), n = 1..20); # G. C. Greubel, Jan 13 2020 MATHEMATICA LinearRecurrence[{34, -1}, {1, 35}, 15] (* Ant King, Nov 08 2011 *) Fibonacci[4*Range -2, 2]/2 (* G. C. Greubel, Jan 13 2020 *) PROG (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 (PARI) vector(21, n, polchebyshev(2*(n-1), 2, 3) ) \\ G. C. Greubel, Jan 13 2020 (Sage) [lucas_number1(4*n-2, 2, -1)/2 for n in (1..20)] # G. C. Greubel, Jan 13 2020 (GAP) List([0..20], n-> Lucas(2, -1, 4*n-2)/2 ); # G. C. Greubel, Jan 13 2020 CROSSREFS Cf. A000129, A008844, A046177, A078522, A284876. Cf. A001109, A001110 (partial sums). Sequence in context: A115492 A187538 A306685 * A162847 A029546 A305539 Adjacent sequences:  A046173 A046174 A046175 * A046177 A046178 A046179 KEYWORD nonn,easy AUTHOR EXTENSIONS Chebyshev comments from Wolfdieter Lang, Nov 29 2002 STATUS approved

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Last modified September 21 13:00 EDT 2020. Contains 337272 sequences. (Running on oeis4.)