A169998 -id:A169998 - cp's OEIS frontend

A107920 Lucas and Lehmer numbers with parameters (1 +- sqrt(-7))/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, 802165, 2209943, 605613, -3814273

Offset: 0

Michael Somos, May 28 2005

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.
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
Row sums of Riordan array (1/(1+2*x^2), x/(1+2*x^2)). - Paul Barry, Sep 10 2005
Pisano period lengths: 1, 1, 8, 2, 24, 8, 21, 2, 24, 24, 10, 8, 168, 21, 24, 4, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012
This is the Lucas Sequence U_n(P, Q) = U_n(1, 2). V_n(1, 2) = A002249(n). - Raphie Frank, Dec 25 2013
Note that (A002249(n)/2)^2 + 7*(a(n)/2)^2 = 2^n for all n in N. This is a specific case of the Lucas sequence identity (V_n/2)^2 - D*(U_n/2)^2 = Q^n where V_n = (a^n + b^n), U_n = (a^n - b^n)/(a - b), Q = (a*b) = 2 and D = (a - b)^2 = -7; a = (1 + sqrt(-7))/2 and b = (1 - sqrt(-7))/2. - Raphie Frank, Nov 26 2015
For the special case where |a(n)| = 1, true for n if and only if n is in {1, 2, 3, 5, 13} = {A215795(n) + 1} = {A060728(n) - 2}, then, additionally, by the Lucas sequence identity (U_2n = U_n*V_n), we have (a(2n)/2)^2 + 7*(a(n)/2)^2 = 2^n. - Raphie Frank, Nov 26 2015

			G.f. = x + x^2 - x^3 - 3*x^4 - x^5 + 5*x^6 + 7*x^7 - 3*x^8 - 17*x^9 - 11*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Christian Ballot, Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions, J. Int. Seq., Vol. 21 (2018), Article 18.6.5.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.
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.
M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Lehmer Number
Wikipedia, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (1,-2).
Index entries for sequences related to Chebyshev polynomials.

Similar sequences: A001607, A077020, A107920, A167433, A169998.

Cf. A002249, A049310, A060728, A109466, A215795, A227078.

[0] cat [n le 2 select 1 else Self(n-1)-2*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 27 2015

a:= n-> (Matrix([[1,1],[ -2,0]])^n)[1,2]: seq(a(n), n=0..45); # Alois P. Heinz, Sep 03 2008

LinearRecurrence[{1, -2}, {0, 1}, 50] (* Vincenzo Librandi, Nov 27 2015 *)
a[ n_] := Im[ ((1 + Sqrt[-7]) / 2)^n // FullSimplify] 2 / Sqrt[7]; (* Michael Somos, Jan 19 2017 *)
a[n_] := If[n < 2, n, Hypergeometric2F1[1 - n/2, (1 - n)/2, 1 - n, 8]];
Table[a[n], {n, 0, 45}] (* Peter Luschny, Feb 23 2018 *)

PARI
```
{a(n) = imag(quadgen(-7)^n)};
```

my(x='x+O('x^100)); concat(0, Vec(x/(1-x+2*x^2))) \\ Altug Alkan, Dec 04 2015

[lucas_number1(n,1,2) for n in range(0, 46)] # Zerinvary Lajos, Apr 22 2009

G.f.: x / (1 - x + 2*x^2).
a(n) = a(n-1) - 2*a(n-2).
a(n) = -(-1)^n*A001607(n).
From Paul Barry, Sep 10 2005: (Start)
a(n+1) = Sum_{k=0..n} C((n+k)/2, k)*(-2)^((n-k)/2)*(1+(-1)^(n-k))/2.
a(n+1) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-2)^k. (End)
a(n+1) = Sum_{k=0..n} A109466(n,k)*2^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = ((1 - i*sqrt(7))^n - (1 + i*sqrt(7))^n)*i/(2^n*sqrt(7)), where i=sqrt(-1). - Bruno Berselli, Jul 01 2011
(a(2*(A060728(n)) - 4))^2 = (A002249(A060728(n) - 2))^2 = 2^(A060728(n)) - 7 = A227078(n), the Ramanujan-Nagell squares. - Raphie Frank, Dec 25 2013
a(n) = -a(-n) * 2^n for all n in Z. - Michael Somos, Jan 19 2017
G.f.: x / (1 - x / (1 + 2*x / (1 - 2*x))). - Michael Somos, Jan 19 2017
a(n) = S(n-1, 1/sqrt(2))*(sqrt(2))^(n-1), n >= 0, with the Chebyshev S polynomials (coefficients in A049310), and S(-1, x) = 0. - Wolfdieter Lang, Feb 22 2018
a(n) = hypergeom([1-n/2, (1-n)/2], [1-n], 8) for n >= 2. - Peter Luschny, Feb 23 2018

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

Original entry on oeis.org

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

A077020 a(n) is the unique odd positive solution x of 2^n = 7x^2+y^2.

Original entry on oeis.org

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, 802165, 2209943, 605613

Offset: 3

Ed Pegg Jr, Oct 17 2002

nonn

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

			G.f. = x^3 + x^4 + x^5 + 3*x^6 + x^7 + 5*x^8 + 7*x^9 + 3*x^10 + 17*x^11 + ...
a(3)=1 since 2^3=8=7*1^2+1^2, a(6)=3 since 2^6=64=7*3^2+1^2.

A. Engel, Problem-Solving Strategies. p. 126.

T. D. Noe, Table of n, a(n) for n=3..500
Eric Weisstein's World of Mathematics, Diophantine Equations 2nd Powers

a(n)=abs(A001607(n-2)).

Cf. A077021.

a(n) = 2^(n-2) * a(4-n) for all n in Z. - Michael Somos, Jan 05 2017
0 = 8*a(n)^2 + 2*a(n+1)^2 - a(n+2)^2 - a(n+3)^2 for all n in Z. - Michael Somos, Jan 05 2017
2*a(n) + a(n+1) = a(n+2) or a(n+3). - Michael Somos, Jan 05 2017

A167433 Row sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

Original entry on oeis.org

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, 802165, 2209943

Offset: 0

Paul Barry, Nov 03 2009

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

Variants are A107920 and A001607.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-2).

a[n_] := Sin[n*ArcTan[Sqrt[7]]]; FullSimplify[Join[{1}, Table[- (2^(n/2 + 1)/Sqrt[7])*(2*a[n] + Sqrt[2]*a[n + 1]), {n, 1, 100}]]] (* or *) Join[{1}, LinearRecurrence[{1,-2},{-3,-1},100]] (* G. C. Greubel, Jun 13 2016 *)

G.f.: (1-4x+4x^2)/(1-x+2x^2).
From G. C. Greubel, Jun 13 2016: (Start)
a(n) = a(n-1) - 2*a(n-2).
a(n) = -(2^((n+2)/2)/sqrt(7))*( 2*sin(n*arctan(sqrt(7))) + sqrt(2)*sin((n+1)*arctan(sqrt(7))) ), n>=1, and a(0)=1. (End)

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A107920 Lucas and Lehmer numbers with parameters (1 +- sqrt(-7))/2.

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

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A077020 a(n) is the unique odd positive solution x of 2^n = 7x^2+y^2.

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A167433 Row sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

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