A172250 -id:A172250 - cp's OEIS frontend

A001607 a(n) = -a(n-1) - 2*a(n-2).

0, 1, -1, -1, 3, -1, -5, 7, 3, -17, 11, 23, -45, -1, 91, -89, -93, 271, -85, -457, 627, 287, -1541, 967, 2115, -4049, -181, 8279, -7917, -8641, 24475, -7193, -41757, 56143, 27371, -139657, 84915, 194399, -364229, -24569, 753027, -703889

Offset: 0

N. J. A. Sloane

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

Apart from the sign, this is an example of a sequence of Lehmer numbers. In this case, the two parameters, alpha and beta, are (1 +- i*sqrt(7))/2. Bilu, Hanrot, Voutier and Mignotte show that all terms of a Lehmer sequence a(n) have a primitive factor for n > 30. Note that for this sequence, a(30) = 24475 = 5*5*11*89 has no primitive factors. - T. D. Noe, Oct 29 2003

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Y. Bilu, G. Hanrot, P. M. Voutier, and M. Mignotte, Existence of primitive divisors of Lucas and Lehmer numbers, [Research Report] RR-3792, INRIA. 1999, pp. 41, HAL Id : inria-00072867.
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 14.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 82.
Erwin Just, Problem E2367, Amer. Math. Monthly, 79 (1972), 772.
G. P. Michon, Never Back to -1.
M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
Eric Weisstein's World of Mathematics, Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (-1,-2).

Apart from signs, same as A077020.
Cf. A001607, A002249, A077021, A107920, A167433, A169998.
Cf. A172250.

[n eq 1 select 0 else n eq 2 select 1 else -Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Aug 22 2011

LinearRecurrence[{-1,-2},{0,1},60] (* Harvey P. Dale, Aug 21 2011 *)

a(n)=if(n<0,0,polcoeff(x/(1+x+2*x^2)+x*O(x^n),n))

a(n)=if(n<0,0,2*imag(((-1+quadgen(-28))/2)^n))

A001607=BinaryRecurrenceSequence(-1,-2,0,1)
[A001607(n) for n in range(51)] # G. C. Greubel, Mar 24 2024

G.f.: x/(1+x+2*x^2).
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n-k-1, k)*2^k = -2/sqrt(7)*(-sqrt(2))^n*(sin(n*arctan(sqrt(7)))). - Vladeta Jovovic, Feb 05 2003
x/(x^2+x+2) = Sum_{n>=0} a(n)*(x/2)^n. - Benoit Cloitre, Mar 12 2002
4*2^n = A002249(n)^2 + 7*A001607(n)^2. See A077020, A077021.
a(n+1) = Sum_{k=0..n} A172250(n,k)*(-1)^k. - Philippe Deléham, Feb 15 2012
G.f.: x - 2*x^2 + 2*x^2/(G(0)+1) where G(k) = 1 + x/(1 - x/(x - 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 16 2012
a(n) = 2^((n-1)/2)*ChebyshevU(n-1, -1/(2*sqrt(2))). - G. C. Greubel, Mar 24 2024
a(n) = (i*(((-1 - i*sqrt(7))/2)^n - ((-1 + i*sqrt(7))/2)^n))/sqrt(7). - Alan Michael Gómez Calderón, Jul 09 2024; after T. D. Noe, Oct 29 2003

A088139 a(n) = 2a(n-1) - 6a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, -2, -16, -20, 56, 232, 128, -1136, -3040, 736, 19712, 35008, -48256, -306560, -323584, 1192192, 4325888, 1498624, -22958080, -54907904, 27932672, 385312768, 603029504, -1105817600, -5829812224, -5024718848, 24929435648, 80007184384

Offset: 0

Paul Barry, Sep 20 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-6).

Cf. A045873, A084102, A088136, A088137.

a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]-6*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 22 2018

seq(coeff(series(x/(1-2*x+6*x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 23 2018

Join[{a=0,b=1},Table[c=2*b-6*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
TrigExpand@Table[(6^(n/2) Sin[n ArcTan[Sqrt[5]]])/Sqrt[5], {n, 0, 20}] (* or *)
Table[Sum[(-5)^k Binomial[n, 2 k + 1], {k, 0, n/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 20 2016 *)
LinearRecurrence[{2,-6},{0,1},40] (* Harvey P. Dale, Nov 22 2024 *)

x='x+O('x^30); concat([0], Vec(x/(1-2*x+6*x^2))) \\ G. C. Greubel, Oct 22 2018

[lucas_number1(n,2,6) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1-2*x+6*x^2).
E.g.f.: exp(x)*sin(sqrt(5)*x)/sqrt(5).
a(n) = ((1+i*sqrt(5))^n-(1-i*sqrt(5))^n)/(2*i*sqrt(5)).
a(n) = Im{(1+i*sqrt(5))^n/sqrt(5)}.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k+1)(-5)^k.
a(n+1) = (-1)^n*Sum_{k, 0<=k<=n} A172250(n,k)*(-2)^k. - Philippe Deléham, Feb 15 2012

cp's OEIS Frontend

A001607 a(n) = -a(n-1) - 2*a(n-2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

SageMath

Formula

A088139 a(n) = 2a(n-1) - 6a(n-2), a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

cp's OEIS Frontend

A001607 a(n) = -a(n-1) - 2*a(n-2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

SageMath

Formula

A088139 a(n) = 2*a(n-1) - 6*a(n-2), a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A088139 a(n) = 2a(n-1) - 6a(n-2), a(0)=0, a(1)=1.