A001542 - OEIS

%I M2030 N0802 #278 Aug 06 2024 23:54:17

%S 0,2,12,70,408,2378,13860,80782,470832,2744210,15994428,93222358,

%T 543339720,3166815962,18457556052,107578520350,627013566048,

%U 3654502875938,21300003689580,124145519261542,723573111879672

%N a(n) = 6*a(n-1) - a(n-2) for n > 1, a(0)=0 and a(1)=2.

%C 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

%C 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

%C Also numbers m such that A125650(6*m^2) is an even perfect square, where A124650(m) is a numerator of m*(m+3)/(4*(m+1)*(m+2)) = Sum_{k=1..m} 1/(k*(k+1)*(k+2)). Sequence A033581 is a bisection of A125651. - _Alexander Adamchuk_, Nov 30 2006

%C 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 = {a(n)}. - _Clark Kimberling_, Aug 26 2008

%C Even Pell numbers. - _Omar E. Pol_, Dec 10 2008

%C Numbers k such that 2*k^2+1 is a square. - _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010

%C These are the integer square roots of the Half-Squares, A007590(k), which occur at values of k given by A001541. Also the numbers produced by adding m + sqrt(floor(m^2/2) + 1) when m is in A002315. See array in A227972. - _Richard R. Forberg_, Aug 31 2013

%C 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

%C Conjecture: Numbers k such that c/m < k 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

%C 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, 0 < a < b < c are given by a=a(n), b=A005319(n), c=a(n+1), x=A001541(n), y=A001653(n+1), z=A002315(n) with 0 < n. - _Michael Somos_, Jun 26 2022

%D Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002; pp. 480-481.

%D Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, pp. 77-79.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D P.-F. Teilhet, Query 2376, L'Intermédiaire des Mathématiciens, 11 (1904), 138-139. - _N. J. A. Sloane_, Mar 08 2022

%H T. D. Noe, <a href="/A001542/b001542.txt">Table of n, a(n) for n = 0..100</a>

%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), 181-193.

%H Christian Aebi and Grant Cairns, <a href="https://arxiv.org/abs/2006.07566">Lattice Equable Parallelograms</a>, arXiv:2006.07566 [math.NT], 2020.

%H Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

%H H. Brocard, <a href="https://gdz.sub.uni-goettingen.de/id/PPN598948236_0004?tify={%22pages%22:[186],%22view%22:%22info%22}">Notes élémentaires sur le problème de Peel</a>, Nouvelle Correspondance Mathématique, 4 (1878), 161-169.

%H A. J. C. Cunningham, <a href="https://archive.org/details/binomialfactoris01cunn/page/n46/mode/1up">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.

%H S. Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, 2014, 5, 2226-2234.

%H R. J. Hetherington, <a href="/A000129/a000129.pdf">Letter to N. J. A. Sloane, Oct 26 1974</a>.

%H J. M. Katri and D. R. Byrkit, <a href="http://www.jstor.org/stable/2313820">Problem E1976</a>, Amer. Math. Monthly, 75 (1968), 683-684.

%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 3, k=1.

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968268">On the multiple solutions of the Pell equation</a>, Annals Math., 30 (1928), 66-72.

%H D. H. Lehmer, <a href="/A001542/a001542.pdf">On the multiple solutions of the Pell equation</a> (annotated scanned copy).

%H Mathematical Reflections, <a href="https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2013-05/mr_4_2013_solutions.pdf">Solution to Problem O271</a>, Issue 5, 2013, p 22.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H B. Polster and M. Ross, <a href="https://arxiv.org/abs/1503.04658">Marching in squares</a>, arXiv preprint arXiv:1503.04658 [math.HO], 2015.

%H Mark A. Shattuck, <a href="https://www.emis.de/journals/INTEGERS/papers/j5/j5.Abstract.html">Tiling proofs of some formulas for the Pell numbers of odd index</a>, Integers, 9 (2009), 53-64.

%H R. A. Sulanke, <a href="https://math.boisestate.edu/~sulanke/PAPERS/cutpasteII.pdf">Moments, Narayana numbers and the cut and paste for lattice paths</a>.

%H Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, <a href="https://hal.archives-ouvertes.fr/hal-02918958/document#page=18">Integer sequences and ellipse chains inside a hyperbola</a>, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).

%F a(n) = 2*A001109(n).

%F a(n) = ((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) / (2*sqrt(2)).

%F G.f.: 2*x/(1-6*x+x^2).

%F a(n) = sqrt(2*(A001541(n))^2 - 2)/2. - _Barry E. Williams_, May 07 2000

%F a(n) = (C^(2n) - C^(-2n))/sqrt(8) where C = sqrt(2) + 1. - _Gary W. Adamson_, May 11 2003

%F For all terms x of the sequence, 2*x^2 + 1 is a square. Limit_{n->oo} a(n)/a(n-1) = 3 + 2*sqrt(2). - _Gregory V. Richardson_, Oct 10 2002

%F 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

%F a(n) = ((1+sqrt(2))^(2*n) - (1-sqrt(2))^(2*n))/(2*sqrt(2)). - _Antonio Alberto Olivares_, Dec 24 2003

%F 2*A001541(k)*A001653(n)*A001653(n+k) = A001653(n)^2 + A001653(n+k)^2 + a(k)^2; e.g., 2*3*5*29 = 5^2 + 29^2 + 2^2; 2*99*29*5741 = 29^2 + 5741^2 + 70^2. - _Charlie Marion_, Oct 12 2007

%F a(n) = sinh(2*n*arcsinh(1))/sqrt(2). - _Herbert Kociemba_, Apr 24 2008

%F For n > 0, a(n) = A001653(n) + A002315(n-1). - _Richard R. Forberg_, Aug 31 2013

%F a(n) = 3*a(n-1) + 2*A001541(n-1); e.g., a(4) = 70 = 3*12 + 2*17. - _Zak Seidov_, Dec 19 2013

%F a(n)^2 + 1^2 = A115598(n)^2 + (A115598(n)+1)^2. - _Hermann Stamm-Wilbrandt_, Jul 27 2014

%F Sum _{n >= 1} 1/( a(n) + 1/a(n) ) = 1/2. - _Peter Bala_, Mar 25 2015

%F E.g.f.: exp(3*x)*sinh(2*sqrt(2)*x)/sqrt(2). - _Ilya Gutkovskiy_, Dec 07 2016

%F A007814(a(n)) = A001511(n). See Mathematical Reflections link. - _Michel Marcus_, Jan 06 2017

%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Jan 20 2017

%F From _A.H.M. Smeets_, May 28 2017: (Start)

%F A051009(n) = a(2^(n-2)).

%F a(2n) = 2*a(2)*A001541(n).

%F A001541(n)/a(n) > sqrt(2) > 2*a(n)/A001541(n). (End)

%F a(A298210(n)) = A002349(2*n^2). - _A.H.M. Smeets_, Jan 25 2018

%F a(n) = A000129(n)*A002203(n). - _Adam Mohamed_, Jul 20 2024

%e G.f. = 2*x + 12*x^2 + 70*x^3 + 408*x^4 + 2378*x^5 + 13860*x^6 + ...

%p A001542:=2*z/(1-6*z+z**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%p seq(combinat:-fibonacci(2*n, 2), n = 0..20); # _Peter Luschny_, Jun 28 2018

%t LinearRecurrence[{6, -1}, {0, 2}, 30] (* _Harvey P. Dale_, Jun 11 2011 *)

%t Fibonacci[2*Range[0,20], 2] (* _G. C. Greubel_, Dec 23 2019 *)

%t Table[2 ChebyshevU[-1 + n, 3], {n, 0, 20}] (* _Herbert Kociemba_, Jun 05 2022 *)

%o (Haskell)

%o a001542 n = a001542_list !! n

%o a001542_list =

%o    0 : 2 : zipWith (-) (map (6 *) $ tail a001542_list) a001542_list

%o -- _Reinhard Zumkeller_, Aug 14 2011

%o (Maxima)

%o a[0]:0$

%o a[1]:2$

%o a[n]:=6*a[n-1]-a[n-2]$

%o A001542(n):=a[n]$

%o makelist(A001542(x),x,0,30); /* _Martin Ettl_, Nov 03 2012 */

%o (PARI) {a(n) = imag( (3 + 2*quadgen(8))^n )}; /* _Michael Somos_, Jan 20 2017 */

%o (PARI) vector(21, n, 2*polchebyshev(n-1, 2, 33) ) \\ _G. C. Greubel_, Dec 23 2019

%o (Python)

%o l=[0, 2]

%o for n in range(2, 51): l+=[6*l[n - 1] - l[n - 2], ]

%o print(l) # _Indranil Ghosh_, Jun 06 2017

%o (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

%o (Sage) [2*chebyshev_U(n-1,3) for n in (0..20)] # _G. C. Greubel_, Dec 23 2019

%o (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

%Y Bisection of Pell numbers A000129: {a(n)} and A001653(n+1), n >= 0.

%Y Cf. A001108, A001353, A001541, A001835, A003499, A007805, A007913, A115598, A125650, A125651, A125652.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_