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 A001542 a(n) = 6*a(n-1) - a(n-2) for n > 1, a(0)=0 and a(1)=2. (Formerly M2030 N0802) 67
 0, 2, 12, 70, 408, 2378, 13860, 80782, 470832, 2744210, 15994428, 93222358, 543339720, 3166815962, 18457556052, 107578520350, 627013566048, 3654502875938, 21300003689580, 124145519261542, 723573111879672 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Consider the equation core(x) = core(2x+1) where core(x) is the smallest number such that x*core(x) is a square: solutions are given by a(n)^2, n > 0. - Benoit Cloitre, Apr 06 2002 Terms > 0 give numbers k which are solutions to the inequality |round(sqrt(2)*k)/k - sqrt(2)| < 1/(2*sqrt(2)*k^2). - Benoit Cloitre, Feb 06 2006 Also numbers n such that A125650(6*n^2) is an even perfect square, where A124650(n) is a numerator of n(n+3)/(4(n+1)(n+2)) = Sum_{k=1..n} 1/(k*(k+1)*(k+2)). Sequence A033581 is a bisection of A125651. - Alexander Adamchuk, Nov 30 2006 The upper principal convergents to 2^(1/2), beginning with 3/2, 17/12, 99/70, 577/408, comprise a strictly decreasing sequence; essentially, numerators = A001541 and denominators = A001542. - Clark Kimberling, Aug 26 2008 Even Pell numbers. - Omar E. Pol, Dec 10 2008 Numbers k such that 2*k^2+1 is a square. - Vladimir Joseph Stephan Orlovsky, Feb 19 2010 These are the integer square roots of the Half-Squares, A007590(n), which occur at values of n given by A001541. Also the numbers produced by adding m + sqrt(floor(m^2/2) + 1) when m = A002315. See array in A227972. - Richard R. Forberg, Aug 31 2013 A001541(n)/a(n) is the closest rational approximation of sqrt(2) with a denominator not larger than a(n), and 2*a(n)/A001541(n) is the closest rational approximation of sqrt(2) with a numerator not larger than 2*a(n). These rational approximations together with those obtained from the sequences A001653 and A002315 give a complete set of closest rational approximations of sqrt(2) with restricted numerator as well as denominator. - A.H.M. Smeets, May 28 2017 Conjecture: Numbers n such that c/m < n for all natural a^2 + b^2 = c^2 (Pythagorean triples), a < b < c and a+b+c = m. Numbers which correspondingly minimize c/m are A002939. - Lorraine Lee, Jan 31 2020 All of the positive integer solutions of a*b+1=x^2, a*c+1=y^2, b*c+1=z^2, x+z=2*y, 0infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson, Oct 10 2002 For n > 0: a(n) = A001652(n) + A046090(n) - A001653(n); e.g. 70 = 119 + 120 - 169. Also a(n) = A001652(n - 1) + A046090(n - 1) + A001653(n - 1); e.g., 70 = 20 + 21 + 29. Also a(n)^2 + 1 = A001653(n - 1)*A001653(n); e.g., 12^2 + 1 = 145 = 5*29. Also a(n + 1)^2 = A084703(n + 1) = A001652(n)*A001652(n + 1) + A046090(n)*A046090(n + 1). - Charlie Marion, Jul 01 2003 a(n) = ((1+sqrt(2))^(2*n)-(1-sqrt(2))^(2*n))/(2*sqrt(2)). - Antonio Alberto Olivares, Dec 24 2003 2*A001541(k)*A001653(n)*A001653(n+k) = A001653(n)^2 + A001653(n+k)^2 + a2(k)^2; e.g., 2*3*5*29 = 5^2+29^2+2^2; 2*99*29*5741 = 2*99*29*5741 = 29^2+5741^2+70^2. - Charlie Marion, Oct 12 2007 a(n) = sinh(2*n*arcsinh(1))/sqrt(2). - Herbert Kociemba, Apr 24 2008 For n > 0, a(n) = A001653(n) + A002315(n-1). - Richard R. Forberg, Aug 31 2013 a(n) = 3*a(n-1) + 2*A001541(n-1); e.g., a(4) = 70 = 3*12+2*17. - Zak Seidov, Dec 19 2013 a(n)^2 + 1^2 = A115598(n)^2 + (A115598(n)+1)^2. - Hermann Stamm-Wilbrandt, Jul 27 2014 Sum _{n >= 1} 1/( a(n) + 1/a(n) ) = 1/2. - Peter Bala, Mar 25 2015 E.g.f.: exp(3*x)*sinh(2*sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Dec 07 2016 A007814(a(n)) = A001511(n). See Mathematical Reflections link. - Michel Marcus, Jan 06 2017 a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 20 2017 From A.H.M. Smeets, May 28 2017: (Start) A051009(n) = a(2^(n-2)). a(2n) =2*a(2)*A001541(n). A001541(n)/a(n) > sqrt(2) > 2*a(n)/A001541(n). (End) a(A298210(n)) = A002349(2*n^2). - A.H.M. Smeets, Jan 25 2018 EXAMPLE G.f. = 2*x + 12*x^2 + 70*x^3 + 408*x^4 + 2378*x^5 + 13860*x^6 + ... MAPLE A001542:=2*z/(1-6*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation seq(combinat:-fibonacci(2*n, 2), n = 0..20); # Peter Luschny, Jun 28 2018 MATHEMATICA LinearRecurrence[{6, -1}, {0, 2}, 30] (* Harvey P. Dale, Jun 11 2011 *) Fibonacci[2*Range[0, 20], 2] (* G. C. Greubel, Dec 23 2019 *) Table[2 ChebyshevU[-1 + n, 3], {n, 0, 20}] (* Herbert Kociemba, Jun 05 2022 *) PROG (Haskell) a001542 n = a001542_list !! n a001542_list =    0 : 2 : zipWith (-) (map (6 *) \$ tail a001542_list) a001542_list -- Reinhard Zumkeller, Aug 14 2011 (Maxima) a:0\$ a:2\$ a[n]:=6*a[n-1]-a[n-2]\$ A001542(n):=a[n]\$ makelist(A001542(x), x, 0, 30); /* Martin Ettl, Nov 03 2012 */ (PARI)  {a(n) = imag( (3 + 2*quadgen(8))^n )}; /* Michael Somos, Jan 20 2017 */ (PARI) vector(21, n, 2*polchebyshev(n-1, 2, 33) ) \\ G. C. Greubel, Dec 23 2019 (Python) l=[0, 2] for n in range(2, 51): l+=[6*l[n - 1] - l[n - 2], ] print(l) # Indranil Ghosh, Jun 06 2017 (Magma) I:=[0, 2]; [n le 2 select I[n] else 6Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 23 2019 (Sage) [2*chebyshev_U(n-1, 3) for n in (0..20)] # G. C. Greubel, Dec 23 2019 (GAP) a:=[0, 2];; for n in [3..20] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019 CROSSREFS Bisection of Pell numbers A000129: {a(n)} and A001653(n+1), n >= 0. Cf. A001108, A001353, A001541, A001835, A003499, A007805, A007913, A115598, A125650, A125651, A125652. Sequence in context: A243771 A026306 A116398 * A059229 A001251 A143357 Adjacent sequences:  A001539 A001540 A001541 * A001543 A001544 A001545 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified October 5 14:03 EDT 2022. Contains 357258 sequences. (Running on oeis4.)