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 A001610 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 2. (Formerly M0764 N0291) 51
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 Odd composite numbers n such that n divides a(n-1) are in A005845. - Zak Seidov, May 04 2010; comment edited by N. J. A. Sloane, Aug 10 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 From Gus Wiseman, Feb 12 2019: (Start) 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) 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 T. D. Noe, Table of n, a(n) for n = 0..500 Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv:1505.06339 [math.NT], 2015. N-N. Cao, F-Z. Zhao, Some Properties of Hyperfibonacci and Hyperlucas Numbers, J. Int. Seq. 13 (2010) # 10.8.8. Ligia L. Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, Journal of Integer Sequences, Vol. 19, 2016, Issue 7, #16.7.6. P. Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2. F. Hazama, Spectra of graphs attached to the space of melodies, Discrete Math., 311 (2011), 2368-2383. See Table 2.1. Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 96. K. Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1. Rui Liu and Feng-Zhen Zhao, On the Sums of Reciprocal Hyperfibonacci Numbers and Hyperlucas Numbers, Journal of Integer Sequences, Vol. 15 (2012), #12.4.5. - From N. J. A. Sloane, Oct 05 2012 R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, sequence a_{1,s}, arXiv:0903.2514 [math.NT], 2009-2011. N. Neumärker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, Journal of Integer Sequences 12 (2009) 09.4.5, Example 10. 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 A. Radhakrishnan, L. Solus, and C. Uhler. Counting Markov equivalence classes for DAG models on trees, Discrete Applied Mathematics 244 (2018): 170-185. V. Shevelev, On divisibility of C(n-i-1,i-1) by i, Int. J. of Number Theory, 3 (2007), no.1, 119-139. [Vladimir Shevelev, Apr 23 2010] J. W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398. Li-Na Zheng, Rui Liu, and Feng-Zhen Zhao, On the Log-Concavity of the Hyperfibonacci Numbers and the Hyperlucas Numbers, Journal of Integer Sequences, Vol. 17 (2014), #14.1.4. Index entries for linear recurrences with constant coefficients, signature (2,0,-1). FORMULA a(n) = A000204(n)-1 = A000032(n+1)-1 = A000071(n+1) + A000045(n). 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) = A014217(n+1) - A000035(n+1). - Paul Curtz, Sep 21 2008 a(n) = -1 + (1/2)*(1/2 + (1/2)*sqrt(5))^n + (1/2)*(1/2 + (1/2)*sqrt(5))^n*sqrt(5) - (1/2)*sqrt(5)*(1/2 - (1/2)*sqrt(5))^n + (1/2)*(1/2 - (1/2)*sqrt(5))^n, with n >= 0. - Paolo P. Lava, Sep 29 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(n) = Sum_{k=0..n-1} Lucas(k), where Lucas(n) = A000032(n). - Gary Detlefs, Dec 07 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 For n > 1, a(n) = A048162(n+1) + 3. - Toby Gottfried, Apr 13 2013 For n > 0, a(n) = A014217(n-1) + A014217(n). - Paolo P. Lava, Mar 20 2015 For n > 0, a(n) = A169985(n + 1) - 1. - Gus Wiseman, Feb 12 2019 MAPLE with(combinat): seq(fibonacci(n)+fibonacci(n+2)-1, n=0..40); # Zerinvary Lajos, Jan 31 2008 g:=(1+z^2)/(1-z-z^2): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-1, n=1..40); # Zerinvary Lajos, Jan 09 2009 MATHEMATICA 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, a==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 *) LinearRecurrence[{2, 0, -1}, {2, 3, 6}, 20] (* Eric W. Weisstein, Feb 13 2018 *) PROG (Haskell) a001610 n = a001610_list !! n a001610_list =    0 : 2 : map (+ 1) (zipWith (+) a001610_list (tail a001610_list)) -- Reinhard Zumkeller, Aug 21 2011 (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) a(n)=([0, 1, 0; 0, 0, 1; -1, 0, 2]^n*[0; 2; 3])[1, 1] \\ Charles R Greathouse IV, Sep 08 2016 (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) -1); # G. C. Greubel, Jul 12 2019 CROSSREFS Cf. A000032 (first differences), A000071, A000126, A000204, A000296, A001644, A032190, A169985, A174625, A306357. Sequence in context: A026647 A026669 A023614 * A324015 A238777 A344615 Adjacent sequences:  A001607 A001608 A001609 * A001611 A001612 A001613 KEYWORD nonn,easy,hear AUTHOR STATUS approved

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Last modified October 6 02:32 EDT 2022. Contains 357261 sequences. (Running on oeis4.)