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A001350 Associated Mersenne numbers.
(Formerly M3250 N1311)
22
0, 1, 1, 4, 5, 11, 16, 29, 45, 76, 121, 199, 320, 521, 841, 1364, 2205, 3571, 5776, 9349, 15125, 24476, 39601, 64079, 103680, 167761, 271441, 439204, 710645, 1149851, 1860496, 3010349, 4870845, 7881196, 12752041, 20633239, 33385280, 54018521, 87403801, 141422324 (list; graph; refs; listen; history; text; internal format)
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) = A050140(Fibonacci(n)). - Thomas Baruchel, Nov 05 2002

a(n) = Lucas_number(n) - 1 - (-1)^n = A000032(n) - 1 - (-1)^n. - Hieronymus Fischer, Feb 18 2006

a(n) = 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

C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22.

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.

M. Baake, J. Hermisson, P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056.

S. Butler, The art of juggling with two balls or a proof for a modular condition of Lucas numbers, arXiv:1004.4293v2 [math.CO], 2010.

J. W. Cannon, W. J. Floyd, L. Lambert, W. R. Parry, and J. S. Purcell, Bitwist manifolds and two-bridge knots, arXiv preprint arXiv:1306.4564 [math.GT], 2013-2015.

J. W. Cannon, W. J. Floyd, L. Lambert, W. R. Parry, and J. 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 [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.

N. Neumarker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, Journal of Integer Sequences, 12 (2009), Article 09.4.5, Example 10.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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.

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).

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) = 2*A074331(n) - A000045(n). - Paul Barry, Jul 19 2004

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) = A052952(n-1) + A052952(n-3). - R. J. Mathar, Jul 02 2018

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

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

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

CROSSREFS

Cf. A031367, A098554, A000032, A004146, A129744.

Sequence in context: A176115 A066898 A118143 * A077238 A185507 A000286

Adjacent sequences:  A001347 A001348 A001349 * A001351 A001352 A001353

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

Additional comments from Michael Somos, Aug 01 2002

STATUS

approved

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Last modified December 7 03:00 EST 2019. Contains 329836 sequences. (Running on oeis4.)