OFFSET
0,4
COMMENTS
a(n) is last term in the period of the continued fraction expansion of phi^n (phi being the golden number). E.g.: n=10, phi^10=[122,1,121,1,121,1,121,...] (and the period may only have 1 or 2 terms). Also, a(n) = floor(phi^n)-((n+1) mod 2), or a(n) = A014217(n)-((n+1) mod 2). - Thomas Baruchel, Nov 05 2002 [continued fraction value corrected by Jon E. Schoenfield, Jan 20 2019]
a(n) is the resultant of the polynomials x^2-x-1 and x^(n+1)-x^n-1 for n >= 1. - Richard Choulet, Aug 05 2007
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - Michael Somos, Feb 12 2012
Gives the number of arrangements of black and white beads on a necklace with a total of n beads satisfying (1) there is at least one black bead (2) between any two black beads the number of white beads is even and (3) rotations and flippings of a necklace are considered distinct (see Butler). - Peter Bala, Mar 06 2014
This is the case P1 = 1, P2 = 0, Q = -1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014
The resultant of the (s_2, s_2+n) pair, where s_n(X) is X^n-X-1, is -a(n). See Rush link. - Michel Marcus, Sep 30 2019
REFERENCES
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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gene Abrams and Gonzalo Aranda Pino, The Leavitt path algebras of generalized Cayley graphs, arXiv preprint arXiv:1310.4735 [math.RA], 2013.
Michael Baake, Joachim Hermisson, and Peter A. B. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056.
Michael Baake, John A. G. Roberts, and Alfred Weiss, Periodic orbits of linear endomorphism of the 2-torus and its lattices, arXiv:0808.3489 [math.DS] (2008).
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 21.
Steve Butler, The art of juggling with two balls or a proof for a modular condition of Lucas numbers, arXiv:1004.4293v2 [math.CO], 2010.
James W. Cannon, William J. Floyd, LeeR Lambert, Walter R. Parry and Jessica S. Purcell, Bitwist manifolds and two-bridge knots, arXiv preprint arXiv:1306.4564 [math.GT], 2013-2015.
James W. Cannon, William J. Floyd, LeeR Lambert, Walter R. Parry and Jessica S. Purcell, Bitwist manifolds and two-bridge knots, Pacific Journal of Mathematics 284 (2016), 1-39.
N. Garnier and O. Ramare, Fibonacci numbers and trigonometric identities, Fibonacci Quart. 46/47 (2008/2009), no. 1, 56-61.
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
Petros Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.
C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22.
C. B. Haselgrove, Associated Mersenne numbers [Annotated and scanned copy]
G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290; see Thm. 7.12.
B. Myers, Number of spanning trees in a wheel, IEE Trans. Circuit Theo. 18 (2) (1971) 280-282, Table 1.
Natascha Neumärker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, Journal of Integer Sequences, 12 (2009), Article 09.4.5, Example 10.
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, J. Int. Seq. (2024) Art. 24.5.7. See p. 2.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
David E. Rush, Degree n Relatives of the Golden Ratio and Resultants of the Corresponding Polynomials, Fib. Q. 50(4), 2012, 313-325. See p. 317.
Jianxin Wei and Yujun Yang, Associated Mersenne graphs, arXiv:2407.08237 [math.CO], 2024. See p. 4.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).
FORMULA
G.f.: x*(1+x^2)/((1-x^2)*(1-x-x^2)). - Simon Plouffe in his 1992 dissertation
a(n) = a(n-1) + a(n-2) + 1 -(-1)^n. a(-n) = (-1)^n * a(n).
a(n) = A050140(Fibonacci(n)). - Thomas Baruchel, Nov 05 2002
Convolution of F(n) and {1, 0, 2, 0, 2, ...}. a(n) = Sum_{k=0..n} ((1+(-1)^k)-0^k)*F(n-k) = Sum_{k=0..n} F(k)*((1+(-1)^(n-k))-0^(n-k)). - Paul Barry, Jul 19 2004
a(n) = Lucas_number(n) - 1 - (-1)^n = A000032(n) - 1 - (-1)^n. - Hieronymus Fischer, Feb 18 2006
a(n) = -(1 - ((1 + sqrt(5))/2)^n - (-(1 + sqrt(5))/2)^(-n) + (-1)^n). - Roger L. Bagula and Gary W. Adamson, Nov 26 2008
a(n) = n * Sum_{k=1..n} (Sum_{i=ceiling((n-k)/2)..(n-k)} (binomial(i,n-k-i)*binomial(k+i-1,k-1))/k*(-1)^(k+1)), n>0. - Vladimir Kruchinin, Sep 03 2010
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). - Colin Barker, Apr 11 2014
a(n) = sqrt(A152152(n)). - Colin Barker, Apr 11 2014
a(n) = a(2*n)/A000032(n) when n is odd; a(n) = a(2*n)/(A000032(n+2)) when n is even. - Bob Selcoe, Jun 01 2014
a(12n+6)/a(4n+2) = (a(6n+3)/a(2n+1))^2. - Bob Selcoe, Jun 01 2014
a(n) = Sum_{k=0..n-1} binomial(k-1, 2*k-n)*n/(n-k). - Peter Luschny, Sep 26 2014
From Peter Bala, Mar 19 2015: (Start)
a(n) = -(alpha^n - 1)*(beta^n - 1), where alpha = 1/2*(1 + sqrt(5)) and beta = (1/2)*(1 - sqrt(5)).
a(n) = -det(I - M^n) where I is the 2 X 2 identity matrix and M = [ 1, 1; 1, 0 ]. Cf. A129744.
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + Sum_{n >= 1} Fibonacci(n)*x^n. Cf. A004146. (End)
a(n) = Sum_{j=n..2*n-1} L(j) mod Sum_{j=0..n-1} L(j) where L(j)=A000032(j). - Art Baker, Jan 20 2019
Convolution of (1, 0, 3, 0, 5, 0, 7, ...) and (1, 1, 1, 2, 3, 5, 8, 13, ...). - Gary W. Adamson, Jul 08 2019
a(n) = Sum_{d|n} d*A060280(d) = Sum_{d|n} A031367(d). [Baake, Roberts, Weiss, eq(2)]. - R. J. Mathar, Oct 19 2021
EXAMPLE
G.f. = x + x^2 + 4*x^3 + 5*x^4 + 11*x^5 + 16*x^6 + 29*x^7 + 45*x^8 + 76*x^9 + ...
n=1: a(9)/a(3) = 76/4 = 19; a(18)/a(6) = 5776/16 = 361 = 19^2. - Bob Selcoe, Jun 01 2014
MAPLE
A001350 := n -> add(binomial(k-1, 2*k-n)*n/(n-k), k=0..n-1);
seq(A001350(n), n=0..39); # Peter Luschny, Sep 26 2014
MATHEMATICA
Clear[f, n]; f[n_] = -(1 - ((1 + Sqrt[5])/2)^n - (-(1 + Sqrt[5])/2)^(-n) + (-1)^n); Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 30}] (* Roger L. Bagula and Gary W. Adamson, Nov 26 2008 *)
a[ n_] := LucasL[n] - 1 - (-1)^n; (* Michael Somos, May 18 2015 *)
a[ n_] := SeriesCoefficient[ x D[ Log[ 1 + x / (1 - x - x^2)], x], {x, 0, n}]; (* Michael Somos, May 18 2015 *)
LinearRecurrence[{1, 2, -1, -1}, {0, 1, 1, 4}, 40] (* Jean-François Alcover, Jan 07 2019 *)
PROG
(PARI) {a(n) = fibonacci(n+1) + fibonacci(n-1) - 1 - (-1)^n};
(PARI) {a(n) = my(w = quadgen(5)); simplify( -(w^n - 1) * ((-1/w)^n - 1))}; /* Michael Somos, Feb 12 2012 */
(Magma) [Floor(-(1 - ((1 + Sqrt(5))/2)^n - (-(1 + Sqrt(5))/2)^(-n) + (-1)^n)): n in [0..40]]; // Vincenzo Librandi, Aug 15 2011
(Python)
from sympy import lucas
def A001350(n): return lucas(n)-((n&1^1)<<1) # Chai Wah Wu, Sep 23 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Additional comments from Michael Somos, Aug 01 2002
STATUS
approved