A206776 a(n) = 3*a(n-1) + 2*a(n-2) for n>1, a(0)=2, a(1)=3.

2, 3, 13, 45, 161, 573, 2041, 7269, 25889, 92205, 328393, 1169589, 4165553, 14835837, 52838617, 188187525, 670239809, 2387094477, 8501763049, 30279478101, 107841960401, 384084837405, 1367938433017, 4871984973861, 17351831787617, 61799465310573, 220102059506953

Offset

0,1

Comments

This is the Lucas sequence V(3,-2).

Inverse binomial transform of this sequence is A072265.

a(n) = A124805(n) - 1 for n>0.

References

Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. Exercise 7.49(c), pages 379, 573.

Links

Bruno Berselli, Table of n, a(n) for n = 0..200

Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.

Wikipedia, Lucas sequence: Specific names.

Index entries for linear recurrences with constant coefficients, signature (3,2).

Formula

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

a(n) = ((3-sqrt(17))^n+(3+sqrt(17))^n)/2^n.

a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 17*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

From Michael Somos, Oct 13 2016: (Start)

a(n) = (-2)^n * a(-n) for all n in Z. -

If c = (3 + sqrt(17))/2, then c^n = (a(n) + sqrt(17)*A007482(n-1)) / 2. (End)

E.g.f.: 2*exp(3*x/2)*cosh(sqrt(17)*x/2). - Stefano Spezia, Oct 21 2022

a(n) = 2*A007482(n)-3*A007482(n-1). - R. J. Mathar, Feb 18 2024

Example

G.f. = 2 + 3*x + 13*x^2 + 45*x^3 + 161*x^4 + 573*x^5 + 2041*x^6 + 7269*x^7 + ...

Maple

A206776 := proc(n)
    option remember ;
    if n <= 1 then
        n+2 ;
    else
        3*procname(n-1)+2*procname(n-2) ;
    end if;
end proc:
seq(A206776(n), n=0..30) ; # R. J. Mathar, Feb 18 2024

Mathematica

RecurrenceTable[{a[n] == 3 a[n - 1] + 2 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]
LinearRecurrence[{3, 2}, {2, 3}, 30] (* Harvey P. Dale, Apr 29 2014 *)
a[ n_] := If[ n < 0, (-2)^n a[ -n], ((3 + Sqrt[17])/2)^n + ((3 - Sqrt[17])/2)^n // Expand]; (* Michael Somos, Oct 13 2016 *)
a[ n_] := If[ n < 0, (-2)^n a[ -n], Boole[n == 0] + SeriesCoefficient[ ((1 + 3*x + Sqrt[1 + 6*x + 17*x^2])/2)^n, {x, 0, n}]]; (* Michael Somos, Oct 13 2016 *)

Prog

(Magma) [n le 1 select n+2 else 3*Self(n)+2*Self(n-1): n in [0..25]];
(Maxima) a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+2*a[n-2]$ makelist(a[n], n, 0, 25);
(PARI) Vec((2-3*x)/(1-3*x-2*x^2) + O(x^30)) \\ Michel Marcus, Jun 26 2015
(PARI) {a(n) = 2 * real(( (3 + quadgen(68)) / 2 )^n)}; /* Michael Somos, Oct 13 2016 */
(PARI) {a(n) = my(w = quadgen(-8)); simplify(w^n * subst(2 * polchebyshev(n), x, -3/4*w))}; /* Michael Somos, Oct 13 2016 */
(PARI) for(n=0, 25, print1(round(((3+sqrt(17))/2)^n+((3-sqrt(17))/2)^n), ", ")) \\ Hugo Pfoertner, Nov 19 2018

Crossrefs

Cf. A189736 (same recurrence but with initial values reversed).

Cf. A007482, A072265, A124805.

Sequence in context: A164133 A226938 A301395 * A275556 A214888 A378137

Adjacent sequences: A206773 A206774 A206775 * A206777 A206778 A206779

Keyword

nonn,easy

Author

Bruno Berselli, Jan 10 2013

Status

approved