A189735 a(1)=3, a(2)=1, a(n) = 3*a(n-1) + a(n-2).

3, 1, 6, 19, 63, 208, 687, 2269, 7494, 24751, 81747, 269992, 891723, 2945161, 9727206, 32126779, 106107543, 350449408, 1157455767, 3822816709, 12625905894, 41700534391, 137727509067, 454883061592, 1502376693843, 4962013143121, 16388416123206, 54127261512739

Offset: 1

Author

Harvey P. Dale, Apr 26 2011

Keywords

• nonn
• easy

Comments

This is one of two sequences of positive integers a(n) such that 13*a(n)^2 + 68*(-1)^n = b(n)^2. For n >= 2, b(n)^2 = 13*a(n-1)*a(n+1) - 153*(-1)^n and in this sequence with the values of b(n) = a(n-1) + a(n+1) = {9, 20, 69, 227, ...}. The other sequence is {8, 27, 89, 294, ...} with offset 0 and signature (3,1). - Klaus Purath, Aug 17 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• Index entries for linear recurrences with constant coefficients, signature (3,1).

Programs

Mathematica

LinearRecurrence[{3,1},{3,1},40]

Maxima

a[1]:3$ a[2]:1$ a[n]:=3*a[n-1]+a[n-2]$ makelist(a[n], n, 1, 28); /* Bruno Berselli, May 24 2011 */

PARI

v=vector(99);v[1]=3;v[2]=1;for(i=3,#v,v[i]=3*v[i-1]+v[i-2]);v \\ Charles R Greathouse IV, May 24 2011

Formula

G.f.: x*(3-8*x)/(1-3*x-x^2). - Bruno Berselli, May 24 2011

a(n) = (a(n-1)*a(n-2) + 51*(-1)^n)/a(n-3), n >= 4; a(n) = (a(n-1)^2 - 17*(-1)^n)/a(n-2), n >= 3. - Klaus Purath, Aug 17 2021

a(n+3) = 3^(n+1) + Sum_{k=0..n} a(k+1)*3^(n-k). - Greg Dresden and Canran Wang, Jun 13 2024