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A001350
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Associated Mersenne numbers.
(Formerly M3250 N1311)
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27
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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
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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) = 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(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
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) = (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
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EXAMPLE
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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
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MAPLE
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A001350 := n -> add(binomial(k-1, 2*k-n)*n/(n-k), k=0..n-1);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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