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%I M1928 N0762 #104 Dec 27 2023 11:20:44
%S 0,2,9,35,132,494,1845,6887,25704,95930,358017,1336139,4986540,
%T 18610022,69453549,259204175,967363152,3610248434,13473630585,
%U 50284273907,187663465044,700369586270,2613814880037,9754889933879,36405744855480,135868089488042
%N a(n) = 4*a(n-1) - a(n-2) + 1, with a(0) = 0, a(1) = 2.
%C Second member of the Diophantine pair (m,k) that satisfies 3(m^2 + m) = k^2 + k: a(n) = k. - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002
%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).
%H T. D. Noe, <a href="/A001571/b001571.txt">Table of n, a(n) for n = 0..200</a>
%H Niccolò Castronuovo, <a href="https://arxiv.org/abs/2102.02739">On the number of fixed points of the map gamma</a>, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
%H Vladimir Pletser, <a href="https://arxiv.org/abs/2101.00998">Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers</a>, arXiv:2101.00998 [math.NT], 2021.
%H Vladimir Pletser, <a href="https://arxiv.org/abs/2102.12392">Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers</a>, arXiv:2102.12392 [math.GM], 2021.
%H Vladimir Pletser, <a href="https://arxiv.org/abs/2102.13494">Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations</a>, arXiv:2102.13494 [math.NT], 2021.
%H Vladimir Pletser, <a href="https://arxiv.org/abs/2103.03019">Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers</a>, arXiv:2103.03019 [math.GM], 2021.
%H Vladimir Pletser, <a href="https://doi.org/10.13140/RG.2.2.35428.91527">Searching for multiple of triangular numbers being triangular numbers</a>, 2021.
%H Vladimir Pletser, <a href="https://www.researchgate.net/profile/Vladimir-Pletser/publication/359808848_USING_PELL_EQUATION_SOLUTIONS_TO_FIND_ALL_TRIANGULAR_NUMBERS_MULTIPLE_OF_OTHER_TRIANGULAR_NUMBERS/">Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers</a>, 2021.
%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 Jamie Radcliffe and Adam Volk, <a href="https://arxiv.org/abs/2101.04213">Generalized saturation problems for cliques, paths, and stars</a>, arXiv:2101.04213 [math.CO], 2021.
%H V. Thebault, <a href="http://www.jstor.org/stable/2305033">Consecutive cubes with difference a square</a>, Amer. Math. Monthly, 56 (1949), 174-175.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).
%F a(n) = (A001834(n) - 1)/2.
%F G.f.: x*(2-x)/( (1-x)*(1-4*x+x^2) ). - _Simon Plouffe_ in his 1992 dissertation.
%F a(n) = sqrt((-2 + (2 - sqrt(3))^n + (2 + sqrt(3))^n)*(2 + (2 - sqrt(3))^(1 + n) + (2 + sqrt(3))^(1 + n)))/(2*sqrt(2)). - _Gerry Martens_, Jun 05 2015
%F E.g.f.: (exp(2*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - exp(x))/2. - _Franck Maminirina Ramaharo_, Nov 12 2018
%F a(n) = 2*A061278(n) - A061278(n-1). - _R. J. Mathar_, Feb 06 2020
%F a(n) = ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n)/4 - (1/2). - _Vladimir Pletser_, Jan 15 2021
%F a(n) = (ChebyshevU(n, 2) + ChebyshevU(n-1, 2) - 1)/2. - _G. C. Greubel_, Feb 02 2022
%p f := gfun:-rectoproc({a(0) = 0, a(1) = 2, a(n) = 4*a(n - 1) - a(n - 2) + 1}, a(n), remember): map(f, [$ (0 .. 40)])[]; # _Vladimir Pletser_, Jul 25 2020
%t a[0]=0; a[1]=2; a[n_]:= a[n]= 4a[n-1] -a[n-2] +1; Table[a[n], {n, 0, 24}] (* _Robert G. Wilson v_, Apr 24 2004 *)
%t Table[(ChebyshevU[n,2] +ChebyshevU[n-1,2] -1)/2, {n,0,30}] (* _G. C. Greubel_, Feb 02 2022 *)
%o (Magma) I:=[0,2]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2)+1: n in [1..30]]; // _Vincenzo Librandi_, Jun 07 2015
%o (Magma) [(Evaluate(ChebyshevU(n+1), 2) + Evaluate(ChebyshevU(n), 2) - 1)/2 : n in [0..30]]; // _G. C. Greubel_, Feb 02 2022
%o (Sage) [(chebyshev_U(n,2) + chebyshev_U(n-1,2) - 1)/2 for n in (0..30)] # _G. C. Greubel_, Feb 02 2022
%Y Cf. A001353, A001834, A061278, A071954, A082840.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Better description from Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002
%E More terms and new description from _Robert G. Wilson v_, Apr 24 2004
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