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%I M5409 #61 Dec 27 2023 08:53:48
%S 1,169,32761,6355441,1232922769,239180661721,46399815451081,
%T 9001325016847969,1746210653453054881,338755865444875798921,
%U 65716891685652451935769,12748738231151130799740241,2473189499951633722697670961,479786014252385791072548426169
%N Square hex numbers.
%C Numbers n of the form n = y^2 = 3*x^2 - 3*x + 1.
%D M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 19.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A006051/b006051.txt">Table of n, a(n) for n = 1..435</a>
%H M. Gardner & N. J. A. Sloane, <a href="/A003154/a003154.pdf">Correspondence, 1973-74</a>
%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2018volume18/FG201808index.html">Integer Sequences and Circle Chains Inside a Circular Segment</a>, Forum Geometricorum, Vol. 18 (2018), 47-55.
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexNumber.html">Hex Number.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195,-195,1).
%F a(n) = A001570(n)^2.
%F a(1 - n) = a(n).
%F G.f.: x * (1 - 26*x + x^2) / ((1 - x) * (1 - 194*x + x^2)). - _Simon Plouffe_ in his 1992 dissertation
%F a(n) = 194*a(n-1) - a(n-2) - 24, a(1)=1, a(2)=169. - _James A. Sellers_, Jul 04 2000
%F a(n+1) = A003215(A001921(n)). - _Joerg Arndt_, Jan 02 2017
%F a(n) = (1/8)*(1 + 7*(ChebyshevU(n-1, 97) - ChebyshevU(n-2, 97))). - _G. C. Greubel_, Oct 07 2022
%e G.f. = x + 169*x^2 + 32761*x^3 + 6355441*x^4 + 1232922769*x^5 + ...
%t Rest@ CoefficientList[Series[x(1-26x+x^2)/((1-x)(1-194x+x^2)), {x,0,20}], x] (* _Michael De Vlieger_, Jan 02 2017 *)
%t LinearRecurrence[{195,-195,1},{1,169,32761},20] (* _Harvey P. Dale_, Nov 03 2017 *)
%o (PARI) {a(n) = sqr( real( (2 + quadgen( 12)) ^ (2*n - 1)) / 2)} /* _Michael Somos_, Feb 15 2011 */
%o (Magma) [(7*Evaluate(ChebyshevSecond(n),97) - 7*Evaluate(ChebyshevU(n-1), 97) + 1)/8: n in [1..30]]; // _G. C. Greubel_, Nov 04 2017; Oct 07 2022
%o (SageMath)
%o def A006051(n): return (7*chebyshev_U(n-1,97) - 7*chebyshev_U(n-2,97) + 1)/8
%o [A006051(n) for n in range(1,31)] # _G. C. Greubel_, Oct 07 2022
%Y Cf. A003500.
%Y Intersection of A000290 and A003215.
%Y Values of x are given by A001922, values of y by A001570.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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