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 A002605 a(n) = 2*(a(n-1) + a(n-2)), a(0) = 0, a(1) = 1. 126
 0, 1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920, 136384, 372608, 1017984, 2781184, 7598336, 20759040, 56714752, 154947584, 423324672, 1156544512, 3159738368, 8632565760, 23584608256, 64434348032, 176037912576, 480944521216, 1313964867584 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Individually, both this sequence and A028859 are convergents to 1 + sqrt(3). Mutually, both sequences are convergents to 2 + sqrt(3) and 1 + sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001 The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1, 2, ..., n + 1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004 The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(4). - Cino Hilliard, Sep 25 2005 The Hankel transform of this sequence is [1, 2, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Nov 21 2007 [1, 3; 1, 1]^n *[1, 0] = [A026150(n), a(n)]. - Gary W. Adamson, Mar 21 2008 (1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3). - Gary W. Adamson, Mar 21 2008 a(n+1) is the number of ways to tile a board of length n using red and blue tiles of length one and two. - Geoffrey Critzer, Feb 07 2009 Starting with offset 1 = INVERT transform of the Jacobsthal sequence, A001045: (1, 1, 3, 5, 11, 21, ...). - Gary W. Adamson, May 12 2009 Starting with "1" = INVERTi transform of A007482: (1, 3, 11, 39, 139, ...). - Gary W. Adamson, Aug 06 2010 An elephant sequence, see A175654. For the corner squares four A vectors, with decimal values 85, 277, 337 and 340, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A026150, without the first leading 1. - Johannes W. Meijer, Aug 15 2010 The sequence 0, 1, -2, 6, -16, 44, -120, 328, -896, ... (with alternating signs) is the Lucas U(-2,-2)-sequence. - R. J. Mathar, Jan 08 2013 a(n+1) counts n-walks (closed) on the graph G(1-vertex;1-loop,1-loop,2-loop,2-loop). - David Neil McGrath, Dec 11 2014 Number of binary strings of length 2*n - 2 in the regular language (00+11+0101+1010)*. - Jeffrey Shallit, Dec 14 2015 For n >= 1, a(n) equals the number of words of length n - 1  over {0, 1, 2, 3} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Dec 17 2015 a(n+1) is the number of compositions of n into parts 1 and 2, both of two kinds. - Gregory L. Simay, Sep 20 2017 Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1, ..., n} that have neutral elements. - J. Devillet, Sep 28 2017 (1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3), for n >= 0; integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 10 2018 Starting with 1, 2, 6, 16, ..., number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>3, 1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third and fourth elements. - Sergey Kitaev, Dec 09 2020 REFERENCES John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..500 A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019. Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021. Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5. M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017. M. Diepenbroek, M. Maus, and A. Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3. Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651. Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019. Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26. Dale Gerdemann Bird Flock, Youtube video, 2011. A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=2. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 476 D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124. Tanya Khovanova, Recursive Sequences Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), (41) and (45), lhs, m=2. D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, Journal of the London Mathematical Society 1.3 (1935): 162-165. See U_n. Alan Prince, Counting parses, Rutgers Optimality Archive, 2010. Index entries for linear recurrences with constant coefficients, signature (2,2). FORMULA a(n) = (-I*sqrt(2))^(n-1)*U(n-1, I/sqrt(2)) where U(n, x) is the Chebyshev U-polynomial. - Wolfdieter Lang G.f.: x/(1 - 2*x - 2*x^2). From Paul Barry, Sep 17 2003: (Start) E.g.f. exp(x)*(sinh(sqrt(3)x)/sqrt(3) + cosh(sqrt(3)x)). a(n) = (1 + sqrt(3))^n(1/2 + sqrt(3)/6) + (1 - sqrt(3))^n*(1/2 - sqrt(3)/6). Binomial transform of 1, 1, 3, 3, 9, 9, ... Binomial transform is A079935. (End) a(n) = Sum_{k=0..floor(n/2)} binomial(n - k, k)*2^(n - k). - Paul Barry, Jul 13 2004 a(n) = A080040(n) - A028860(n+1). - Creighton Dement, Jan 19 2005 a(n) = Sum_{k=0..n} A112899(n,k). - Philippe Deléham, Nov 21 2007 a(n) = Sum_{k=0..n} A063967(n,k). - Philippe Deléham, Nov 03 2006 a(n) = ((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*sqrt(3)). a(n) = Sum_{k=0..n} binomial(n, 2*k + 1) * 3^k. Binomial transform of expansion of sinh(sqrt(3)x)/sqrt(3) (0, 1, 0, 3, 0, 9, ...). E.g.f.: exp(x)*sinh(sqrt(3)*x)/sqrt(3). - Paul Barry, May 09 2003 a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/2)*sin(2*Pi*k/3)*(1 + 2*cos(Pi*k/6))^n, n >= 1. - Herbert Kociemba, Jun 02 2004 a(n+1) = ((3 + sqrt(3))*(1 + sqrt(3))^n + (3 - sqrt(3))*(1 - sqrt(3))^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009 Antidiagonals sums of A081577. - J. M. Bergot, Dec 15 2012 G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k + 2 + 2*x)/(x*(4*k + 4 + 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013 a(n) = 2^(n - 1)*hypergeom([1 - n/2, (1 - n)/2], [1 - n], -2)) for n >= 3. - Peter Luschny, Dec 16 2015 Sum_{k=0..n} a(k)*2^(n-k) = a(n+2)/2 - 2^n. - Greg Dresden, Feb 11 2022 a(n) = 2^floor(n/2) * A002530(n). - Gregory L. Simay, Sep 22 2022 MAPLE a:=0:a:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+2*a[n-2]od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 15 2008 a := n -> `if`(n<3, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -2)); seq(simplify(a(n)), n=0..29); # Peter Luschny, Dec 16 2015 MATHEMATICA Expand[Table[((1 + Sqrt)^n - (1 - Sqrt)^n)/(2Sqrt), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *) a[n_]:=(MatrixPower[{{1, 3}, {1, 1}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *) LinearRecurrence[{2, 2}, {0, 1}, 30] (* Robert G. Wilson v, Apr 13 2013 *) Round@Table[Fibonacci[n, Sqrt] 2^((n - 1)/2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *) nxt[{a_, b_}]:={b, 2(a+b)}; NestList[nxt, {0, 1}, 30][[All, 1]] (* Harvey P. Dale, Sep 17 2022 *) PROG (Sage) [lucas_number1(n, 2, -2) for n in range(0, 30)] # Zerinvary Lajos, Apr 22 2009 (Sage) a = BinaryRecurrenceSequence(2, 2) print([a(n) for n in (0..29)])  # Peter Luschny, Aug 29 2016 (PARI) Vec(x/(1-2*x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 10 2011 (PARI) A002605(n)=([2, 2; 1, 0]^n)[2, 1] \\ M. F. Hasler, Aug 06 2018 (Magma) [Floor(((1 + Sqrt(3))^n - (1 - Sqrt(3))^n)/(2*Sqrt(3))): n in [0..30]]; // Vincenzo Librandi, Aug 18 2011 (Haskell) a002605 n = a002605_list !! n a002605_list =    0 : 1 : map (* 2) (zipWith (+) a002605_list (tail a002605_list)) -- Reinhard Zumkeller, Oct 15 2011 (Magma) [n le 2 select n-1 else 2*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 07 2018 CROSSREFS First differences are given by A026150. a(n) = A073387(n, 0), n>=0 (first column of triangle). Equals (1/3) A083337. First differences of A077846. Pairwise sums of A028860 and abs(A077917). a(n) = A028860(n)/2 apart from the initial terms. Row sums of A081577 and row sums of triangle A156710. The following sequences (and others) belong to the same family: A001333, A000129, A026150, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519. Cf. A080953, A052948, A080040, A028859, A030195, A106435, A108898, A125145, A265106, A265107, A265278, A270810, A293005, A293006, A293007. Cf. A175289 (Pisano periods). Cf. A002530. Sequence in context: A335686 A118041 A105073 * A026134 A105696 A074413 Adjacent sequences:  A002602 A002603 A002604 * A002606 A002607 A002608 KEYWORD nonn,easy,changed AUTHOR EXTENSIONS Edited by N. J. A. Sloane, Apr 15 2009 STATUS approved

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Last modified October 4 16:41 EDT 2022. Contains 357239 sequences. (Running on oeis4.)