A142881 a(0) = 0, a(1) = 1, after which, if n=3k: a(n) = 2*a(n-1) - a(n-2), if n=3k+1: a(n) = a(n-1) + a(n-2), if n=3k+2: a(n) = 2*a(n-1) + a(n-2).

0, 1, 2, 3, 5, 13, 21, 34, 89, 144, 233, 610, 987, 1597, 4181, 6765, 10946, 28657, 46368, 75025, 196418, 317811, 514229, 1346269, 2178309, 3524578, 9227465, 14930352, 24157817, 63245986, 102334155, 165580141, 433494437, 701408733, 1134903170

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 28 2008

Keywords

• nonn
• easy

Comments

The original name of the sequence was: A modulo three switched recursion (third kind): a(n)=If[Mod[n, 3] ==2, 2*a(n - 1) + a(n - 2), If[Mod[n, 3] == 1, a(n - 1) + a(n - 2), 2*a(n - 1) - a(n - 2)]].

How is this related to A000045 ? - Antti Karttunen, Jan 29 2016

Links

• Antti Karttunen, Table of n, a(n) for n = 0..120
• Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-1).

Crossrefs

Cf. A000045, A119016, A002965, A002530, A048788.

Programs

Mathematica

Clear[a, n]; a[0] = 0; a[1] = 1; a[n_] := a[n] = If[Mod[n, 3] == 2, 2*a[n - 1] + a[n - 2], If[Mod[n, 3] == 1, a[n - 1] + a[n - 2], 2*a[n - 1] - a[n - 2]]]; b = Table[a[n], {n, 0, 50}]

PARI

a=vector(100); a[1]=1; a[2]=2; for(n=3, #a, if(n%3==0, a[n]=2*a[n-1]-a[n-2], if(n%3==1, a[n]=a[n-1]+a[n-2], a[n]=2*a[n-1]+a[n-2]))); concat(0, a) \\ Colin Barker, Jan 30 2016

Formula

a(n) = If[Mod[n, 3] == 2, 2*a(n - 1) + a(n - 2), If[Mod[n, 3] == 1, a(n - 1) + a(n - 2), 2*a(n - 1) - a(n - 2)]].

a(n) = 7*a(n-3)-a(n-6). G.f.: -x^2*(x^4+2*x^3-3*x^2-2*x-1) / (x^6-7*x^3+1). [Colin Barker, Jan 08 2013]

a(0) = 0, a(1) = 1, after which, if n is a multiple of 3, a(n) = 2*a(n-1) - a(n-2), else, if n is of the form 3k+1, a(n) = a(n-1) + a(n-2), and otherwise [when n is of the form 3k+2], a(n) = 2*a(n-1) + a(n-2). - Antti Karttunen, Jan 29 2016, after the original name of the sequence.

Extensions

Offset corrected and sequence edited by Antti Karttunen, Jan 29 2016