A136161 a(n) = 2*a(n-3) - a(n-6), starting a(0..5) = 0, 5, 2, 1, 3, 1.

0, 5, 2, 1, 3, 1, 2, 1, 0, 3, -1, -1, 4, -3, -2, 5, -5, -3, 6, -7, -4, 7, -9, -5, 8, -11, -6, 9, -13, -7, 10, -15, -8, 11, -17, -9, 12, -19, -10, 13, -21, -11, 14, -23, -12, 15, -25, -13, 16, -27, -14

Offset: 0

Author

Paul Curtz, Mar 16 2008

Keywords

• sign
• easy

Comments

Consider the general recurrence a(n) = k*a(n-1) + (5-2*k)*a(n-2) + (2-k)*a(n-3). The coefficients, in k, can be used to form the triple (k, 5-2*k, 2-k). Each triple is associated with a sequence, for example (0, 5, 2) leads to A111108, A112685, ..., (1, 3, 1) leads to A051927, A097075, ..., and so on. This sequence is formed from the triples {(0, 5, 2), (1, 3, 1), (2, 1, 0), (3, -1, -1), (4, -3, -2), ...}, for k >= 0. (Comment modified by G. C. Greubel, Dec 31 2023).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A097076, A099254, A112685, A135138, A135997, A137241.

Cf. A051927, A097075, A111108, A112685.

Programs

Magma

I:=[0,5,2,1,3,1]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..60]]; // G. C. Greubel, Dec 26 2023

Mathematica

LinearRecurrence[{0,0,2,0,0,-1},{0,5,2,1,3,1},60] (* Harvey P. Dale, Aug 16 2012 *)
Table[PadRight[{n, 5-2*n, 2-n}], {n,0,20}]//Flatten (* _G. C. Greubel, Dec 26 2023 *)

PARI

Vec(x*(5+2*x+x^2-7*x^3-3*x^4)/((1-x)^2*(1+x+x^2)^2+O(x^99))) \\ Charles R Greathouse IV, Jul 06 2011

SageMath

def a(n): # a = A136161
    if n<6: return (0,5,2,1,3,1)[n]
    else: return 2*a(n-3) - a(n-6)
[a(n) for n in range(61)] # G. C. Greubel, Dec 26 2023

Formula

G.f.: x*(5+2*x+x^2-7*x^3-3*x^4) / ( (1-x)^2*(1+x+x^2)^2 ). - R. J. Mathar, Jul 06 2011

a(3n) = n.

a(3n+1) = 5 - 2*n.

a(3n+3) = 2 - n.

a(n) = (1/9)*( 27 - 2*(n+1) - 34*ChebyshevU(n, -1/2) + (-1)^n*(9*A099254(n) - 6*A099254(n-1)) ). - G. C. Greubel, Dec 26 2023