A002249 a(n) = a(n-1) - 2*a(n-2) with a(0) = 2, a(1) = 1.

2, 1, -3, -5, 1, 11, 9, -13, -31, -5, 57, 67, -47, -181, -87, 275, 449, -101, -999, -797, 1201, 2795, 393, -5197, -5983, 4411, 16377, 7555, -25199, -40309, 10089, 90707, 70529, -110885, -251943, -30173, 473713, 534059, -413367, -1481485

Offset

0,1

Comments

4*2^n = A002249(n)^2 + 7*A001607(n)^2. See A077020, A077021.

Among presented initial elements of the sequence a(n), the maximal increasing or decreasing subsequences have length either 3 or 4. - Roman Witula, Aug 21 2012

This is the Lucas Sequence V_n(P, Q) = V_n(1, 2). U_n(1, 2) = A107920(n). - Raphie Frank, Dec 25 2013

The only numbers that occur more than once are 1=a(1)=a(4) and -5=a(3)=a(9). See Noam D. Elkies's posting in the Mathematics Stack Exchange link. - Robert Israel, Dec 21 2016

Links

T. D. Noe, Table of n, a(n) for n = 0..500

Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Mathematics Stack Exchange, An exotic sequence, March 2014.

Wikipedia, Lucas Sequence.

Index entries for linear recurrences with constant coefficients, signature (1,-2).

Formula

G.f.: (2-x)/(1-x+2x^2). - Michael Somos, Oct 18 2002

a(n) = trace(A^n) for the square matrix A=[1, -2; 1, 0]. - Paul Barry, Sep 05 2003

a(n) = 2^((n+2)/2)*cos(-n*acot(sqrt(7)/7)). - Paul Barry, Sep 06 2003

a(n) = (-1)^n*(2*A110512(n) - A001607(n)) = ((1 + i*sqrt(7))/2)^n + ((1 - i*sqrt(7))/2)^n. - Roman Witula, Aug 21 2012

G.f.: G(0), where G(k) = 1 + 1/(1 - x*(7*k+1)/(x*(7*k+8) + 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013

(a(A060728(n) - 2))^2 = (A107920(2*(A060728(n)) - 4))^2 = 2^(A060728(n)) - 7 = A227078(n), the Ramanujan-Nagell squares. - Raphie Frank, Dec 25 2013

a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x - 7*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

a(n) = (A107920(n+1) + 2*A107920(n+2) - A107920(n+3))/2. - Raphie Frank, Nov 28 2015

V_n(P,Q) = a(k*n) = ((a(k) + sqrt(-7))/2)^n + ((a(k) - sqrt(-7))/2)^n for k is in {1, 2, 3, 5, 13} = (A060728(n) - 2), P is in {1, -3, -5, 11, -181} = a(k), and Q is in {2, 4, 8, 32, 8192} = 2^k = (2*A076046(n) + 2) = (A227078(n) - 7)/4. P^2 - 4*Q = -7. - Raphie Frank, Dec 05 2015

From Peter Bala, Nov 16 2022: (Start)

The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all positive integers n and k and all primes p.

A268924(n) == a(3^n) (mod 3^n). (End)

Example

We have a(2)-a(7) = a(5)-a(4) = a(6)+a(4) = a(11)-a(10) = a(12)+a(10)=10. Further the following relations: ((1+i*sqrt(7))/2)^4 + ((1-i*sqrt(7))/2)^4 = 1 and ((1+i*sqrt(7))/2)^8 + ((1-i*sqrt(7))/2)^8 = -31. - Roman Witula, Aug 21 2012
G.f. = 2 + x - 3*x^2 - 5*x^3 + x^4 + 11*x^5 + 9*x^6 - 13*x^7 - 31*x^8 + ...
From Raphie Frank, Dec 05 2015: (Start)
V_n(1, 2) = a(1*n) = ((a(1) + sqrt(-7))/2)^n + ((a(1) - sqrt(-7))/2)^n; a(1) = 1.
V_n(-3, 4) = a(2*n) = ((a(2) + sqrt(-7))/2)^n + ((a(2) - sqrt(-7))/2)^n; a(2) = -3.
V_n(-5, 8) = a(3*n) = ((a(3) + sqrt(-7))/2)^n + ((a(3) - sqrt(-7))/2)^n; a(3) = -5.
V_n(11, 32) = a(5*n) = ((a(5) + sqrt(-7))/2)^n + ((a(5) - sqrt(-7))/2)^n; a(5) = 11.
V_n(-181, 8192) = a(13*n) = ((a(13) + sqrt(-7))/2)^n + ((a(13) - sqrt(-7))/2)^n; a(13) = -181.
(End)

Maple

A002249 := proc(n) option remember; >if n = 1 then 1 elif n = 2 then -3 else A002249(n-1>)-2*A002249(n-2); fi; end;

Mathematica

LinearRecurrence[{1, -2}, {2, 1}, 50] (* Roman Witula, Aug 21 2012 *)
a[ n_] := 2^(n/2) ChebyshevT[ n, 8^(-1/2)] 2; (* Michael Somos, Jun 02 2014 *)
a[ n_] := 2^Min[0, n] SeriesCoefficient[ (2 - x) / (1 - x + 2 x^2), {x, 0, Abs @ n}]; (* Michael Somos, Jun 02 2014 *)
Table[2 Re[((1 + I Sqrt[7])/2)^n], {n, 0, 40}] (* Jean-François Alcover, Jun 02 2017 *)

Prog

(PARI) {a(n) = if( n<0, 2^n * a(-n), polsym(2 - x + x^2, n)[n+1])}; /* Michael Somos, Jun 02 2014 */
(PARI) {a(n) = 2 * real( ((1 + quadgen(-28)) / 2)^n )}; /* Michael Somos, Jun 02 2014 */
(PARI) x='x+O('x^100); Vec((2-x)/(1-x+2*x^2)) \\ Altug Alkan, Dec 04 2015
(Sage) [lucas_number2(n, 1, 2) for n in range(0, 40)] # Zerinvary Lajos, Apr 30 2009
(Magma) I:=[2, 1]; [n le 2 select I[n] else Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Nov 29 2015
(Python)
from sympy import sqrt, re, I
def a(n): return 2*re(((1 + I*sqrt(7))/2)**n)
print([a(n) for n in range(40)]) # Indranil Ghosh, Jun 02 2017

Crossrefs

Cf. A014551, A107920, A060728.

Sequence in context: A367196 A058169 A178074 * A157127 A066748 A106583

Adjacent sequences: A002246 A002247 A002248 * A002250 A002251 A002252

Keyword

sign,easy

Author

N. J. A. Sloane, Iwan Duursma

Status

approved