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%I M2225 N0883 #82 Mar 24 2024 03:24:41
%S 0,1,-1,-1,3,-1,-5,7,3,-17,11,23,-45,-1,91,-89,-93,271,-85,-457,627,
%T 287,-1541,967,2115,-4049,-181,8279,-7917,-8641,24475,-7193,-41757,
%U 56143,27371,-139657,84915,194399,-364229,-24569,753027,-703889
%N a(n) = -a(n-1) - 2*a(n-2).
%C The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.
%C Apart from the sign, this is an example of a sequence of Lehmer numbers. In this case, the two parameters, alpha and beta, are (1 +- i*sqrt(7))/2. Bilu, Hanrot, Voutier and Mignotte show that all terms of a Lehmer sequence a(n) have a primitive factor for n > 30. Note that for this sequence, a(30) = 24475 = 5*5*11*89 has no primitive factors. - _T. D. Noe_, Oct 29 2003
%C The sequence a(n) is connected with the sequence A110512 (see Witula's comments to this one). - _Roman Witula_, Jul 27 2012
%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="/A001607/b001607.txt">Table of n, a(n) for n = 0..500</a>
%H Y. Bilu, G. Hanrot, P. M. Voutier and M. Mignotte, <a href="https://hal.inria.fr/inria-00072867">Existence of primitive divisors of Lucas and Lehmer numbers</a>, [Research Report] RR-3792, INRIA. 1999, pp. 41, HAL Id : inria-00072867.
%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See p. 14.
%H Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 82.
%H Erwin Just, <a href="http://www.jstor.org/stable/2316274">Problem E2367</a>, Amer. Math. Monthly, 79 (1972), 772.
%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#m">Never Back to -1</a>.
%H M. Mignotte, <a href="http://pmb.univ-fcomte.fr/1989/Mignotte.pdf">Propriétés arithmétiques des suites récurrentes</a>, Besançon, 1988-1989, see p. 14. In French.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LehmerNumber.html">Lehmer Number</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-2).
%F G.f.: x/(1+x+2*x^2).
%F a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n-k-1, k)*2^k = -2/sqrt(7)*(-sqrt(2))^n*(sin(n*arctan(sqrt(7)))). - _Vladeta Jovovic_, Feb 05 2003
%F x/(x^2+x+2) = Sum_{n>=0} a(n)*(x/2)^n. - _Benoit Cloitre_, Mar 12 2002
%F 4*2^n = A002249(n)^2 + 7*A001607(n)^2. See A077020, A077021.
%F a(n+1) = Sum_{k=0..n} A172250(n,k)*(-1)^k. - _Philippe Deléham_, Feb 15 2012
%F G.f.: x - 2*x^2 + 2*x^2/(G(0)+1) where G(k) = 1 + x/(1 - x/(x - 1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 16 2012
%F a(n) = 2^((n-1)/2)*ChebyshevU(n-1, -1/(2*sqrt(2))). - _G. C. Greubel_, Mar 24 2024
%t LinearRecurrence[{-1,-2},{0,1},60] (* _Harvey P. Dale_, Aug 21 2011 *)
%o (PARI) a(n)=if(n<0,0,polcoeff(x/(1+x+2*x^2)+x*O(x^n),n))
%o (PARI) a(n)=if(n<0,0,2*imag(((-1+quadgen(-28))/2)^n))
%o (Magma) [n eq 1 select 0 else n eq 2 select 1 else -Self(n-1)-2*Self(n-2): n in [1..50]]; // _Vincenzo Librandi_, Aug 22 2011
%o (SageMath)
%o A001607=BinaryRecurrenceSequence(-1,-2,0,1)
%o [A001607(n) for n in range(51)] # _G. C. Greubel_, Mar 24 2024
%Y Apart from signs, same as A077020.
%Y Cf. A001607, A002249, A077021, A107920, A110512, A167433, A169998.
%Y Cf. A172250.
%K sign,easy
%O 0,5
%A _N. J. A. Sloane_
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