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%I #43 Oct 08 2022 08:34:15
%S 1,7,91,1261,17557,244531,3405871,47437657,660721321,9202660831,
%T 128176530307,1785268763461,24865586158141,346332937450507,
%U 4823795538148951,67186804596634801,935791468814738257,13033893758809700791,181538721154521072811
%N Star numbers (A003154) that are also triangular numbers (A000217).
%C From _Colin Barker_, Jan 06 2015: (Start)
%C Also indices of centered square numbers (A001844) which are also centered triangular numbers (A005448).
%C Also indices of centered octagonal numbers (A016754) which are also centered hexagonal numbers (A003215).
%C Also positive integers y in the solutions to 3*x^2-4*y^2-3*x+4*y = 0, the corresponding values of x being A001922.
%C (End)
%H Colin Barker, <a href="/A156712/b156712.txt">Table of n, a(n) for n = 1..875</a>
%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2016volume16/FG2016volume16.pdf#page=423">Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences</a>, Forum Geometricorum, Volume 16 (2016) 419-427.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Star_number">Star Numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%F a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).
%F If x^2 - 3*y^2 = 1 with x even then a(y) = (y+2)/4 evidently related to A001570 by: add 1 and halve.
%F G.f.: x*(1 - 8*x + x^2)/((1-x)*(1 - 14*x + x^2)). - _Alexander R. Povolotsky_, Feb 15 2009
%F a(n) = (4 + (2 + sqrt(3))*(7 - 4*sqrt(3))^n + (2 - sqrt(3))*(7 + 4*sqrt(3))^n)/8. - _Colin Barker_, Mar 05 2016
%F a(n) = (1/2)*( 1 + ChebyshevU(n, 7) - 13*ChebyshevU(n-1, 7) ). - _G. C. Greubel_, Oct 07 2022
%p f:= gfun[rectoproc]({a(n+3)=15*a(n+2)-15*a(n+1)+a(n),a(1)=1,a(2)=7,a(3)=91},a(n),'remember'):
%p seq(f(n), n=1..30); # _Robert Israel_, Jan 01 2015
%t f[n_] := (Simplify[(2 + Sqrt@3)^(2 n - 1) + (2 - Sqrt@3)^(2 n - 1)] + 4)/8; Array[f, 17] (* _Robert G. Wilson v_, Oct 28 2010 *)
%o (PARI) Vec(-x*(x^2-8*x+1)/((x-1)*(x^2-14*x+1)) + O(x^100)) \\ _Colin Barker_, Jan 01 2015
%o (Magma) [(Evaluate(ChebyshevSecond(n+1),7) - 13*Evaluate(ChebyshevU(n), 7) + 1)/2: n in [1..30]]; // _G. C. Greubel_, Oct 07 2022
%o (SageMath)
%o def A156712(n): return (1 + chebyshev_U(n, 7) - 13*chebyshev_U(n-1, 7))/2
%o [A156712(n) for n in range(1,31)] # _G. C. Greubel_, Oct 07 2022
%Y Cf. A000217, A001570, A001844, A001922, A003154.
%Y Cf. A003215, A005448, A007655, A016754.
%K easy,nonn
%O 1,2
%A _Aaron Meyerowitz_, Feb 14 2009
%E a(11) onwards from _Robert G. Wilson v_, Oct 28 2010
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