A098150 a(n) = 2(a(n-2) - a(n-1)) + a(n-3) where a(0)=-3, a(1)=11 & a(2)=-30.

-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, -2225563057071, 5826599727518, -15254236125483

Offset: 0

Author

Creighton Dement, Aug 29 2004

Keywords

• easy
• sign

Comments

Sequence relates bisections of Lucas and Fibonacci numbers.

Pisano period lengths: 1, 3, 4, 6, 5, 12, 8, 6, 12, 15, 10, 12, 7, 24, 20, 12, 9, 12, 18, 30, ... - R. J. Mathar, Aug 10 2012

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..2383
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (-3,-1).

Crossrefs

Cf. A098149, A001519, A005248.

Programs

Magma

I:=[-3,11]; [n le 2 select I[n] else -3*Self(n-1)-Self(n-2): n in [1..35]]; // Vincenzo Librandi, Dec 26 2018

Mathematica

a[0] = -3; a[1] = 11; a[2] = -30; a[n_] := a[n] = 2(a[n - 2] - a[n - 1]) + a[n - 3]; Table[ a[n], {n, 0, 25}] (* Robert G. Wilson v, Sep 04 2004 *)
RecurrenceTable[{a[0]==-3,a[1]==11,a[2]==-30,a[n]==2(a[n-2]-a[n-1])+ a[n-3]},a,{n,30}] (* or *) LinearRecurrence[{-3,-1},{-3,11},30] (* Harvey P. Dale, Feb 05 2012 *)
Table[(-1)^(n+1)(3LucasL[2n+1]-Fibonacci[2n]), {n,0,20}] (* Rigoberto Florez, Dec 24 2018 *)

Formula

2*A098149(n) + a(n) = 8*(-1)^(n+1)*A001519(n) - (-1)^(n+1)*A005248(n+1).

a(n) = - 3a(n-1) - a(n-2). - Tanya Khovanova, Feb 02 2007

G.f.: (2x-3)/(1+3x+x^2). - Philippe Deléham, Nov 16 2008

a(n) = (-1)^(n+1)*(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

Extensions

More terms from Robert G. Wilson v, Sep 04 2004