A098150 -id:A098150 - cp's OEIS frontend

A098149 a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n>1.

-1, -1, 4, -11, 29, -76, 199, -521, 1364, -3571, 9349, -24476, 64079, -167761, 439204, -1149851, 3010349, -7881196, 20633239, -54018521, 141422324, -370248451, 969323029, -2537720636, 6643838879, -17393796001, 45537549124, -119218851371

Offset: 0

Creighton Dement, Aug 29 2004

Sequence relates bisections of Lucas and Fibonacci numbers.
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.
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
From Wolfdieter Lang, Oct 12 2020: (Start)
[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.
This binary indefinite quadratic form of discriminant 5, representing 5, has only this family of proper solutions (modulo sign flip), and no improper ones.
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)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Tanya Khovanova, Recursive Sequences
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
Index entries for linear recurrences with constant coefficients, signature (-3,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A098150, A001519, A005248, A000045, A001906, A049310, A027941.

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 *)
LinearRecurrence[{-3,-1},{-1,-1},30] (* Harvey P. Dale, Apr 19 2014 *)
CoefficientList[Series[-(1 + 4 x)/(1 + 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)

G.f.: -(1+4*x)/(1+3*x+x^2). - Philippe Deléham, Nov 19 2006
a(n) = (-1)^n*A002878(n-1). - R. J. Mathar, Jan 30 2011
-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
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

Simpler definition from Philippe Deléham, Nov 19 2006

A167375 a(n)=3*a(n-1)-a(n-2) with a(0)=1, a(1)=3, a(2)=11.

Original entry on oeis.org

1, 3, 11, 30, 79, 207, 542, 1419, 3715, 9726, 25463, 66663, 174526, 456915, 1196219, 3131742, 8199007, 21465279, 56196830, 147125211, 385178803, 1008411198, 2640054791, 6911753175, 18095204734, 47373861027, 124026378347, 324705274014, 850089443695

Offset: 0

Jamel Ghanouchi, Nov 02 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1).

Cf. A135929, A138034.

I:=[1,3,11]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jun 26 2014

Join[{1},LinearRecurrence[{3,-1},{3,11},30]] (* Harvey P. Dale, Jun 25 2014 *)
CoefficientList[Series[(3 x^2 + 1)/(1 - 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 26 2014 *)
Table[3LucasL[2n+1]-Fibonacci[2n], {n,0,20}] (* Rigoberto Florez, Dec 24 2018 *)

a(n) = (-1)^n*A098150(n-1), n>0.
G.f.: (3*x^2+1)/(1-3*x+x^2).
a(n) = 3*L(2n+1)-F(2n), where F(n) is the n-th Fibonacci number and L(n) is the n-th Lucas number. - Rigoberto Florez, Dec 24 2018

Edited by R. J. Mathar, Nov 03 2009

cp's OEIS Frontend

A098149 a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n>1.

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