%I M3250 N1311 #226 Jul 16 2024 15:58:02
%S 0,1,1,4,5,11,16,29,45,76,121,199,320,521,841,1364,2205,3571,5776,
%T 9349,15125,24476,39601,64079,103680,167761,271441,439204,710645,
%U 1149851,1860496,3010349,4870845,7881196,12752041,20633239,33385280,54018521,87403801,141422324
%N Associated Mersenne numbers.
%C 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]
%C 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
%C This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - _Michael Somos_, Feb 12 2012
%C 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
%C 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
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A001350/b001350.txt">Table of n, a(n) for n = 0..1000</a>
%H Gene Abrams and Gonzalo Aranda Pino, <a href="http://arxiv.org/abs/1310.4735">The Leavitt path algebras of generalized Cayley graphs</a>, arXiv preprint arXiv:1310.4735 [math.RA], 2013.
%H Michael Baake, Joachim Hermisson, and Peter A. B. Pleasants, <a href="http://dx.doi.org/10.1088/0305-4470/30/9/016">The torus parametrization of quasiperiodic LI-classes</a>, J. Phys. A 30 (1997), no. 9, 3029-3056.
%H Michael Baake, John A. G. Roberts, and Alfred Weiss, <a href="https://arxiv.org/abs/0808.3489">Periodic orbits of linear endomorphism of the 2-torus and its lattices</a>, arXiv:0808.3489 [math.DS] (2008).
%H Joshua P. Bowman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Bowman/bowman4.html">Compositions with an Odd Number of Parts, and Other Congruences</a>, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 21.
%H Steve Butler, <a href="http://arxiv.org/abs/1004.4293">The art of juggling with two balls or a proof for a modular condition of Lucas numbers</a>, arXiv:1004.4293v2 [math.CO], 2010.
%H James W. Cannon, William J. Floyd, LeeR Lambert, Walter R. Parry and Jessica S. Purcell, <a href="http://arxiv.org/abs/1306.4564">Bitwist manifolds and two-bridge knots</a>, arXiv preprint arXiv:1306.4564 [math.GT], 2013-2015.
%H James W. Cannon, William J. Floyd, LeeR Lambert, Walter R. Parry and Jessica S. Purcell, <a href="http://dx.doi.org/10.2140/pjm.2016.284.1">Bitwist manifolds and two-bridge knots</a>, Pacific Journal of Mathematics 284 (2016), 1-39.
%H N. Garnier and O. Ramare, <a href="http://www.fq.math.ca/Papers1/46_47-1/Ramare_Garnier_11-08.pdf">Fibonacci numbers and trigonometric identities</a>, Fibonacci Quart. 46/47 (2008/2009), no. 1, 56-61.
%H R. K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, June-August 1968</a>
%H Petros Hadjicostas, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Hadjicostas/hadji2.html">Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence</a>, Journal of Integer Sequences, 19 (2016), Article 16.8.2.
%H C. B. Haselgrove, <a href="https://www.archim.org.uk/eureka/archive/Eureka-11.pdf">Associated Mersenne numbers</a>, Eureka, 11 (1949), 19-22.
%H C. B. Haselgrove, <a href="/A001350/a001350.pdf">Associated Mersenne numbers</a> [Annotated and scanned copy]
%H G. I. Lehrer and G. B. Segal, <a href="http://dx.doi.org/10.1007/PL00004831">Homology stability for classical regular semisimple varieties</a>, Math. Zeit., 236 (2001), 251-290; see Thm. 7.12.
%H B. Myers, <a href="http://doi.org/10.1109/TCT.1971.1083273">Number of spanning trees in a wheel</a>, IEE Trans. Circuit Theo. 18 (2) (1971) 280-282, Table 1.
%H Natascha Neumärker, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Neumarker/neumarker3.html">Realizability of Integer Sequences as Differences of Fixed Point Count Sequences</a>, Journal of Integer Sequences, 12 (2009), Article 09.4.5, Example 10.
%H Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Petojevic/peto5.pdf">The Golden Ratio, Factorials, and the Lambert W Function</a>, J. Int. Seq. (2024) Art. 24.5.7. See p. 2.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H David E. Rush, <a href="https://www.fq.math.ca/Papers1/50-4/Rush.pdf">Degree n Relatives of the Golden Ratio and Resultants of the Corresponding Polynomials</a>, Fib. Q. 50(4), 2012, 313-325. See p. 317.
%H Jianxin Wei and Yujun Yang, <a href="https://arxiv.org/abs/2407.08237">Associated Mersenne graphs</a>, arXiv:2407.08237 [math.CO], 2024. See p. 4.
%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012) The John Selfridge Memorial Volume
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).
%F G.f.: x*(1+x^2)/((1-x^2)*(1-x-x^2)). - _Simon Plouffe_ in his 1992 dissertation
%F a(n) = a(n-1) + a(n-2) + 1 -(-1)^n. a(-n) = (-1)^n * a(n).
%F a(n) = A050140(Fibonacci(n)). - _Thomas Baruchel_, Nov 05 2002
%F 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
%F a(n) = 2*A074331(n) - A000045(n). - _Paul Barry_, Jul 19 2004
%F a(n) = Lucas_number(n) - 1 - (-1)^n = A000032(n) - 1 - (-1)^n. - _Hieronymus Fischer_, Feb 18 2006
%F 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
%F 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
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). - _Colin Barker_, Apr 11 2014
%F a(n) = sqrt(A152152(n)). - _Colin Barker_, Apr 11 2014
%F 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
%F a(12n+6)/a(4n+2) = (a(6n+3)/a(2n+1))^2. - _Bob Selcoe_, Jun 01 2014
%F a(n) = Sum_{k=0..n-1} binomial(k-1, 2*k-n)*n/(n-k). - _Peter Luschny_, Sep 26 2014
%F From _Peter Bala_, Mar 19 2015: (Start)
%F a(n) = -(alpha^n - 1)*(beta^n - 1), where alpha = 1/2*(1 + sqrt(5)) and beta = (1/2)*(1 - sqrt(5)).
%F a(n) = -det(I - M^n) where I is the 2 X 2 identity matrix and M = [ 1, 1; 1, 0 ]. Cf. A129744.
%F exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + Sum_{n >= 1} Fibonacci(n)*x^n. Cf. A004146. (End)
%F a(n) = A052952(n-1) + A052952(n-3). - _R. J. Mathar_, Jul 02 2018
%F a(n) = (L(2*n+1) - L(n+1)) mod (L(n+1)-1) for n > 0 where L(k)=A000032(k). - _Art Baker_, Jan 17 2019
%F 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
%F Convolution of (1, 0, 3, 0, 5, 0, 7, ...) and (1, 1, 1, 2, 3, 5, 8, 13, ...). - _Gary W. Adamson_, Jul 08 2019
%F a(n) = Sum_{d|n} d*A060280(d) = Sum_{d|n} A031367(d). [Baake, Roberts, Weiss, eq(2)]. - _R. J. Mathar_, Oct 19 2021
%e 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 + ...
%e n=1: a(9)/a(3) = 76/4 = 19; a(18)/a(6) = 5776/16 = 361 = 19^2. - _Bob Selcoe_, Jun 01 2014
%p A001350 := n -> add(binomial(k-1, 2*k-n)*n/(n-k), k=0..n-1);
%p seq(A001350(n), n=0..39); # _Peter Luschny_, Sep 26 2014
%t 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 *)
%t a[ n_] := LucasL[n] - 1 - (-1)^n; (* _Michael Somos_, May 18 2015 *)
%t a[ n_] := SeriesCoefficient[ x D[ Log[ 1 + x / (1 - x - x^2)], x], {x, 0, n}]; (* _Michael Somos_, May 18 2015 *)
%t LinearRecurrence[{1, 2, -1, -1}, {0, 1, 1, 4}, 40] (* _Jean-François Alcover_, Jan 07 2019 *)
%o (PARI) {a(n) = fibonacci(n+1) + fibonacci(n-1) - 1 - (-1)^n};
%o (PARI) {a(n) = my(w = quadgen(5)); simplify( -(w^n - 1) * ((-1/w)^n - 1))}; /* _Michael Somos_, Feb 12 2012 */
%o (Magma) [Floor(-(1 - ((1 + Sqrt(5))/2)^n - (-(1 + Sqrt(5))/2)^(-n) + (-1)^n)): n in [0..40]]; // _Vincenzo Librandi_, Aug 15 2011
%o (Python)
%o from sympy import lucas
%o def A001350(n): return lucas(n)-((n&1^1)<<1) # _Chai Wah Wu_, Sep 23 2023
%Y Cf. A031367, A098554, A000032, A004146, A129744.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, _R. K. Guy_
%E Additional comments from _Michael Somos_, Aug 01 2002