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A001610
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a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 2.
(Formerly M0764 N0291)
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52
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0, 2, 3, 6, 10, 17, 28, 46, 75, 122, 198, 321, 520, 842, 1363, 2206, 3570, 5777, 9348, 15126, 24475, 39602, 64078, 103681, 167760, 271442, 439203, 710646, 1149850, 1860497, 3010348, 4870846, 7881195, 12752042, 20633238, 33385281, 54018520
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graph;
refs;
listen;
history;
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internal format)
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OFFSET
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0,2
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COMMENTS
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For prime p, p divides a(p-1). - T. D. Noe, Apr 11 2009 [This result follows immediately from the fact that A032190(n) = (1/n)*Sum_{d|n} a(d-1)*phi(n/d). - Petros Hadjicostas, Sep 11 2017]
Generalization. If a(0,x)=0, a(1,x)=2 and, for n>=2, a(n,x)=a(n-1,x)+x*a(n-2,x)+1, then we obtain a sequence of polynomials Q_n(x)=a(n,x) of degree floor((n-1)/2), such that p is prime iff all coefficients of Q_(p-1)(x) are multiple of p (sf. A174625). Thus a(n) is the sum of coefficients of Q_(n-1)(x). - Vladimir Shevelev, Apr 23 2010
a(n) is the number of ways to modify a circular arrangement of n objects by swapping one or more adjacent pairs. E.g., for 1234, new arrangements are 2134, 2143, 1324, 4321, 1243, 4231 (taking 4 and 1 to be adjacent) and a(4) = 6. - Toby Gottfried, Aug 21 2011
For n>2, a(n) equals the number of Markov equivalence classes with skeleton the cycle on n+1 nodes. See Theorem 2.1 in the article by A. Radhakrishnan et al. below. - Liam Solus, Aug 23 2018
For n > 0, also the number of nonempty subsets of {1, ..., n + 1} containing no two cyclically successive elements (cyclically successive means 1 succeeds n + 1). For example, the a(5) = 17 stable subsets are:
{1}, {2}, {3}, {4}, {5}, {6},
{1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},
{1,3,5}, {2,4,6}.
(End)
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REFERENCES
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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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Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 96.
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FORMULA
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G.f.: x*(2-x)/((1-x-x^2)*(1-x)) = (2*x-x^2)/(1-2*x+x^3). [Simon Plouffe in his 1992 dissertation]
a(n) = F(n) + F(n+2) - 1 where F(n) is the n-th Fibonacci number. - Zerinvary Lajos, Jan 31 2008
a(n) = Sum_{i=1..floor((n+1)/2)} ((n+1)/i)*C(n-i,i-1). In more general case of polynomials Q_n(x)=a(n,x) (see our comment) we have Q_n(x) = Sum_{i=1..floor((n+1)/2)}((n+1)/i)*C(n-i,i-1)*x^(i-1). - Vladimir Shevelev, Apr 23 2010
a(0)=0, a(1)=2, a(2)=3; for n>=3, a(n) = 2*a(n-1) - a(n-3). - George F. Johnson, Jan 28 2013
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MATHEMATICA
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t = {0, 2}; Do[AppendTo[t, t[[-1]] + t[[-2]] + 1], {n, 2, 40}]; t
RecurrenceTable[{a[n] == a[n - 1] +a[n - 2] +1, a[0] == 0, a[1] == 2}, a, {n, 0, 40}] (* Robert G. Wilson v, Apr 13 2013 *)
CoefficientList[Series[x (2 - x)/((1 - x - x^2) (1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
Table[Fibonacci[n] + Fibonacci[n + 2] - 1, {n, 0, 40}] (* Eric W. Weisstein, Feb 13 2018 *)
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PROG
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(Haskell)
a001610 n = a001610_list !! n
a001610_list =
0 : 2 : map (+ 1) (zipWith (+) a001610_list (tail a001610_list))
(Magma) I:=[0, 2]; [n le 2 select I[n] else Self(n-1)+Self(n-2)+1: n in [1..40]]; // Vincenzo Librandi, Mar 20 2015
(Magma) [Lucas(n+1) -1: n in [0..40]]; // G. C. Greubel, Jul 12 2019
(PARI) vector(40, n, f=fibonacci; f(n+1)+f(n-1)-1) \\ G. C. Greubel, Jul 12 2019
(Sage) [lucas_number2(n+1, 1, -1) -1 for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) List([0..40], n-> Lucas(1, -1, n+1)[2] -1); # G. C. Greubel, Jul 12 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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