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%I A001921 M4455 N1885 #72 Dec 30 2023 10:56:05
%S A001921 0,7,104,1455,20272,282359,3932760,54776287,762935264,10626317415,
%T A001921 148005508552,2061450802319,28712305723920,399910829332567,
%U A001921 5570039304932024,77580639439715775,1080558912851088832,15050244140475527879,209622859053806301480
%N A001921 a(n) = 14*a(n-1) - a(n-2) + 6 for n>1, a(0)=0, a(1)=7.
%C A001921 (a(n)+1)^3 - a(n)^3 is a square (that of A001570(n)).
%C A001921 The ratio A001570(n)/a(n) tends to sqrt(3) = 1.73205... as n increases. - _Pierre CAMI_, Apr 21 2005
%C A001921 Define a(1)=0 a(2)=7 such that 3*(a(1)^2) + 3*a(1) + 1 = j(1)^2 = 1^2 and 3*(a(2)^2) + 3*a(2) + 1 = j(2)^2 = 13^2. Then a(n) = a(n-2) + 8*sqrt(3*(a(n-1)^2) + 3*a(n-1) + 1). Another definition : a(n) such that 3*(a(n)^2) + 3*a(n) + 1 = j(n)^2. - _Pierre CAMI_, Mar 30 2005
%C A001921 a(n) = A001353(n)*A001075(n+1). For n>0, the triple {a(n), a(n)+1=A001922(n), A001570(n)} forms a near-isosceles triangle with angle 2*Pi/3 bounded by the consecutive sides. - _Lekraj Beedassy_, Jul 21 2006
%C A001921 Numbers n such that A003215(n) is a square, cf. A006051. - _Joerg Arndt_, Jan 02 2017
%D A001921 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
%D A001921 E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185. - _N. J. A. Sloane_, Mar 03 2022
%D A001921 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001921 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001921 T. D. Noe, <a href="/A001921/b001921.txt">Table of n, a(n) for n = 0..200</a>
%H A001921 J. Brenner and E. P. Starke, <a href="http://www.jstor.org/stable/2306254">Problem E702</a>, Amer. Math. Monthly, 53 (1946), 465.
%H A001921 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 A001921 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001921 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexNumber.html">Hex Number</a>
%H A001921 <a href="/https/oeis.org/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%F A001921 G.f.: x*(-7 + x)/(x - 1)/(x^2 - 14*x + 1) (see _Simon Plouffe_ in Maple section).
%F A001921 a(n) = (A028230(n+1)-1)/2. - _R. J. Mathar_, Mar 19 2009
%F A001921 a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3). - _Colin Barker_, Jan 06 2015
%F A001921 a(n) = -1 - a(-1-n) for all n in Z. - _Michael Somos_, Aug 17 2018
%e A001921 G.f. = 7*x + 104*x^2 + 1455*x^3 + 20272*x^4 + 282359*x^5 + 3932760*x^6 + ... - _Michael Somos_, Aug 17 2018
%p A001921 A001921:=z*(-7+z)/(z-1)/(z**2-14*z+1); # Conjectured by _Simon Plouffe_ in his 1992 dissertation.
%t A001921 t = {0, 7}; Do[AppendTo[t, 14*t[[-1]] - t[[-2]] + 6], {20}]; t (* _T. D. Noe_, Aug 17 2012 *)
%t A001921 LinearRecurrence[{15, -15, 1}, {0, 7, 104}, 19] (* _Michael De Vlieger_, Jan 02 2017 *)
%t A001921 a[ n_] := -1/2 + (ChebyshevT[n + 1, 7] - ChebyshevT[n, 7]) / 12; (* _Michael Somos_, Aug 17 2018 *)
%o A001921 (PARI) concat(0, Vec(x*(x-7)/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ _Colin Barker_, Jan 06 2015
%o A001921 (PARI) {a(n) = -1/2 + (polchebyshev(n + 1, 1, 7) - polchebyshev(n, 1, 7)) / 12}; /* _Michael Somos_, Aug 17 2018 */
%o A001921 (Magma) [Round(-1/2 - (1/6)*Sqrt(3)*(7-4*Sqrt(3))^n + (1/6)*Sqrt(3)*(7+4*Sqrt(3))^n + (1/4)*(7+4*Sqrt(3))^n + (1/4)*(7-4*Sqrt(3))^n): n in [0..50]]; // _G. C. Greubel_, Nov 04 2017
%Y A001921 Cf. A001570, A001922, A006051.
%Y A001921 Cf. numbers m such that k*A000217(m)+1 is a square: A006451 for k=1; A233450 for k=3; A001652 for k=4; A129556 for k=5; this sequence for k=6. - _Bruno Berselli_, Dec 16 2013
%K A001921 nonn,easy
%O A001921 0,2
%A A001921 _N. J. A. Sloane_
%E A001921 More terms from _James A. Sellers_, Jul 04 2000

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