A080206 a(1)=a(2)=a(3)=1; a(n)=|a(n-2)+a(n-3)|-2*a(n-1).

1, 1, 1, 0, 2, -3, 8, -15, 35, -63, 146, -264, 611, -1104, 2555, -4617, 10685, -19308, 44684, -80745, 186866, -337671, 781463, -1412121, 3268034, -5905410, 13666733, -24696090, 57153503, -103277649, 239012711, -431901276, 999537614, -1806186663, 4180009664, -7553370279

Offset

1,5

Links

Table of n, a(n) for n=1..36.

Formula

For n>4, a(2n)+2a(2n-1)+a(2n-2)+a(2n-3)=0.

Let c=4.181943336... be the largest positive root of 1-x-4*x^2+x^3. Then a(2n) is asymptotic to a*c^n, where a=-0.0493553440330328..., while a(2n+1) is asymptotic to b*c^n, where b= 0.11422175750398... (-a/b is the smallest root (in absolute value) of 3-12*x+13*x^2-3*x^3.)

limit n ->infinity a(2n+1)/a(2n)=-2.3142733525986128..., the largest root (in absolute value) of 3+13*x+12*x^2+3*x^3; limit n ->infinity a(2n)/a(2n-1)=-1.8070222... the largest root (in absolute value) of -3+3*x+8*x^2+3*x^3.

Empirical G.f.: x*(2*x^8-2*x^7-4*x^5-3*x^4-4*x^3-3*x^2+x+1) / (x^6-x^4-4*x^2+1). [Colin Barker, Dec 01 2012]

Mathematica

nxt[{a_, b_, c_}]:={b, c, Abs[a+b]-2c}; NestList[nxt, {1, 1, 1}, 40][[All, 1]] (* Harvey P. Dale, Apr 28 2018 *)

Crossrefs

Sequence in context: A331829 A166920 A242510 * A132862 A055543 A308433

Adjacent sequences: A080203 A080204 A080205 * A080207 A080208 A080209

Keyword

sign,easy

Author

Benoit Cloitre, Feb 05 2003

Status

approved