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%I M4569 N1946 #81 Oct 07 2022 09:09:44
%S 1,8,105,1456,20273,282360,3932761,54776288,762935265,10626317416,
%T 148005508553,2061450802320,28712305723921,399910829332568,
%U 5570039304932025,77580639439715776,1080558912851088833,15050244140475527880,209622859053806301481
%N Numbers k such that 3*k^2 - 3*k + 1 is both a square (A000290) and a centered hexagonal number (A003215).
%C Also larger of two consecutive integers whose cubes differ by a square. Defined by a(n)^3 - (a(n) - 1)^3 = square.
%C Let m be the n-th ratio 2/1, 7/4, 26/15, 97/56, 362/209, ... Then a(n) = m*(2-m)/(m^2-3). The numerators 2, 7, 26, ... of m are A001075. The denominators 1, 4, 15, ... of m are A001353.
%C From _Colin Barker_, Jan 06 2015: (Start)
%C Also indices of centered triangular numbers (A005448) which are also centered square numbers (A001844).
%C Also indices of centered hexagonal numbers (A003215) which are also centered octagonal numbers (A016754).
%C Also positive integers x in the solutions to 3*x^2 - 4*y^2 - 3*x + 4*y = 0, the corresponding values of y being A156712.
%C (End)
%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 Vincenzo Librandi, <a href="/A001922/b001922.txt">Table of n, a(n) for n = 0..800</a>
%H J. Brenner and E. P. Starke, <a href="http://www.jstor.org/stable/2306254">Problem E702</a>, Amer. Math. Monthly, 53 (1946), 465.
%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 Sociedad Magic Penny Patagonia, <a href="http://www.magicpenny.org/engteorema.htm">Leonardo en Patagonia</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%F a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).
%F a(n) = (s1*t1^n + s2*t2^n + 6)/12 where s1 = 3 + 2*sqrt(3), s2 = 3 - 2*sqrt(3), t1 = 7 + 4*sqrt(3), t2 = 7 - 4*sqrt(3).
%F a(n) = A001075(n)*A001353(n+1).
%F G.f.: (1-7*x)/((1-x)*(1-14*x+x^2)). - _Simon Plouffe_ (in his 1992 dissertation) and _Colin Barker_, Jan 01 2012
%F a(n) = A076139(n+1) - 7*A076139(n). - _R. J. Mathar_, Jul 14 2015
%F a(n) = (1/2)*(1 + ChebyshevU(n, 7) + ChebyshevU(n-1, 7)). _G. C. Greubel_, Oct 07 2022
%e 8 is in the sequence because 3*8^2 - 3*8 + 1 = 169 is a square and also a centered hexagonal number. - _Colin Barker_, Jan 07 2015
%p seq(simplify((1 +ChebyshevU(n,7) +ChebyshevU(n-1,7))/2), n=0..30); # _G. C. Greubel_, Oct 07 2022
%t With[{s1=3+2Sqrt[3],s2=3-2Sqrt[3],t1=7+4Sqrt[3],t2=7-4Sqrt[3]}, Simplify[ Table[(s1 t1^n+s2 t2^n+6)/12,{n,0,20}]]] (* or *) LinearRecurrence[ {15,-15,1},{1,8,105},21] (* _Harvey P. Dale_, Aug 14 2011 *)
%t CoefficientList[Series[(1-7*x)/(1-15*x+15*x^2-x^3),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 16 2012 *)
%o (Magma) I:=[1, 8, 105]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Apr 16 2012
%o (PARI) Vec((1-7*x)/(1-15*x+15*x^2-x^3) + O(x^100)) \\ _Colin Barker_, Jan 06 2015
%o (SageMath) [(1+chebyshev_U(n,7) +chebyshev_U(n-1,7))/2 for n in range(30)] # _G. C. Greubel_, Oct 07 2022
%Y Cf. A001075, A001353, A001570, A001844, A001921, A003215.
%Y Cf. A005448, A006051, A016754, A076139, A156712.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Additional comments from _James R. Buddenhagen_, Mar 04 2001
%E Name improved by _Colin Barker_, Jan 07 2015
%E Edited by _Robert Israel_, Feb 20 2017
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