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%I #69 Dec 18 2023 12:18:20
%S -1,-1,4,-11,29,-76,199,-521,1364,-3571,9349,-24476,64079,-167761,
%T 439204,-1149851,3010349,-7881196,20633239,-54018521,141422324,
%U -370248451,969323029,-2537720636,6643838879,-17393796001,45537549124,-119218851371
%N a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n>1.
%C Sequence relates bisections of Lucas and Fibonacci numbers.
%C 2*a(n) + A098150(n) = 8*(-1)^(n+1)*A001519(n) - (-1)^(n+1)*A005248(n+1). Apparently, if (z(n)) is any sequence of integers (not all zero) satisfying the formula z(n) = 2(z(n-2) - z(n-1)) + z(n-3) then |z(n+1)/z(n)| -> golden ratio phi + 1 = (3+sqrt(5))/2.
%C Pisano period lengths: 1, 3, 4, 6, 1, 12, 8, 6, 12, 3, 10, 12, 7, 24, 4, 12, 9, 12, 18, 6, ... . - _R. J. Mathar_, Aug 10 2012
%C From _Wolfdieter Lang_, Oct 12 2020: (Start)
%C [X(n) = (-1)^n*(S(n, 3) + S(n-1, 3)), Y(n) = X(n-1)] gives all integer solutions (modulo sign flip between X and Y) of X^2 + Y^2 + 3*X*Y = +5, for n = -oo..+oo, with Chebyshev S polynomials (A049310), with S(-1, x) = 0, S(-|n|, x) = - S(|n|-2, x), for |n| >= 2, and S(n,-x) = (-1)^n*S(n, x). The present sequence is a(n) = -X(n-1), for n >= 0. See the formula section.
%C This binary indefinite quadratic form of discriminant 5, representing 5, has only this family of proper solutions (modulo sign flip), and no improper ones.
%C This comment is inspired by a paper by Robert K. Moniot (private communication) See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)
%H Vincenzo Librandi, <a href="/A098149/b098149.txt">Table of n, a(n) for n = 0..1000</a>
%H Seong Ju Kim, R. Stees, and L. Taalman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Stees/stees4.html">Sequences of Spiral Knot Determinants</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Ryan Stees, <a href="https://commons.lib.jmu.edu/honors201019/84">Sequences of Spiral Knot Determinants</a>, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F G.f.: -(1+4*x)/(1+3*x+x^2). - _Philippe Deléham_, Nov 19 2006
%F a(n) = (-1)^n*A002878(n-1). - _R. J. Mathar_, Jan 30 2011
%F -a(n+1) = Sum_{k, 0<=k<=n}(-5)^k*Binomial(n+k, n-k) = Sum_{k, 0<=k<=n}(-5)^k*A085478(n, k). - _Philippe Deléham_, Nov 28 2006
%F a(n) = (-1)^n*(S(n-1, 3) + S(n-2, 3)) = (-1)^n*S(2*(n-1), sqrt(5)), for n >= 0, with Chebyshev S polynomials (A049310), with S(-1, x) = 0 and S(-2, x) = -1. S(n, 3) = A001906(n+1) = F(2*(n+1)), with F = A000045. - _Wolfdieter Lang_, Oct 12 2020
%t a[0] = a[1] = -1; a[n_] := a[n] = -3a[n - 2] - a[n - 1]; Table[ a[n], {n, 0, 27}] (* _Robert G. Wilson v_, Sep 01 2004 *)
%t LinearRecurrence[{-3,-1},{-1,-1},30] (* _Harvey P. Dale_, Apr 19 2014 *)
%t CoefficientList[Series[-(1 + 4 x)/(1 + 3 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Apr 19 2014 *)
%Y Cf. A098150, A001519, A005248, A000045, A001906, A049310, A027941.
%K easy,sign
%O 0,3
%A _Creighton Dement_, Aug 29 2004
%E Simpler definition from _Philippe Deléham_, Nov 19 2006
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