A201455 a(n) = 3*a(n-1) + 4*a(n-2) for n>1, a(0)=2, a(1)=3.

2, 3, 17, 63, 257, 1023, 4097, 16383, 65537, 262143, 1048577, 4194303, 16777217, 67108863, 268435457, 1073741823, 4294967297, 17179869183, 68719476737, 274877906943, 1099511627777, 4398046511103, 17592186044417, 70368744177663, 281474976710657

Offset: 0

Author

Bruno Berselli, Jan 09 2013

Keywords

• nonn
• easy

Comments

This is the Lucas sequence V(3,-4).

Inverse binomial transform of this sequence is A087451.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..200
• Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
• Wikipedia, Lucas sequence: Specific names.
• Index entries for linear recurrences with constant coefficients, signature (3,4).

Crossrefs

Cf. for the same recurrence with initial values (i,i+1): A015521 (Lucas sequence U(3,-4); i=0), A122117 (i=1), A189738 (i=3).

Cf. for similar closed form: A014551 (2^n+(-1)^n), A102345 (3^n+(-1)^n), A087404 (5^n+(-1)^n).

Cf. A052539, A024036.

Programs

Magma

[n le 1 select n+2 else 3*Self(n)+4*Self(n-1): n in [0..25]];

Mathematica

RecurrenceTable[{a[n] == 3 a[n - 1] + 4 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]

Maxima

a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+4*a[n-2]$ makelist(a[n], n, 0, 25);

PARI

Vec((2-3*x)/((1+x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Jun 26 2015

Formula

G.f.: (2-3*x)/((1+x)*(1-4*x)).

a(n) = 4^n+(-1)^n.

a(n) = A086341(A047524(n)) for n>0, a(0)=2.

a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 25*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

a(n) = (2/4^n) * Sum_{k = 0..n} binomial(4*n+1, 4*k). - Peter Bala, Feb 06 2019