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 A078057 Expansion of (1+x)/(1-2*x-x^2). 67
 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243, 275807, 665857, 1607521, 3880899, 9369319, 22619537, 54608393, 131836323, 318281039, 768398401, 1855077841, 4478554083, 10812186007, 26102926097, 63018038201, 152139002499, 367296043199, 886731088897 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Let x_n be the sequence 1,3,7,17,41,99,239,... (this sequence or A001333) and let y_n = 1,2,5,12,29,70,169,... (A000129). Then {+- x_n +- y_n*sqrt(2) } are the units in the ring of algebraic integers Z[ sqrt(2) ]. Consider a string of n red, blue and green beads (with start and end points distinct and not interchangeable). If one pairing is disallowed, so that a red bead cannot immediately follow a blue bead or vice versa, how many different strings exist of any given length? Answer is a(n). E.g., a(3)=17 because there are 17 strings of length 3: RRR, RRG, RGR, RGG, RGB, GRR, GRG, GGR, GGG, GGB, GBG, GBB, BGR, BGG, BGB, BBG, BBB - Wayne VanWeerthuizen, May 02 2004 The number of Khalimsky-continuous functions with one fixed endpoint. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007 The sequence (-1)^C(n+1,2)*a(n) with g.f. (1-3x-x^2-x^3)/(1+6x^2+x^4) is the Hankel transform of the signed central binomial coefficients (-1)^C(n+1,2)*A001405(n). - Paul Barry, Jun 24 2008 An elephant sequence, see A175655. For the central square six A vectors, with decimal values between 21 and 336, lead to this sequence. For the corner squares these vectors lead to the companion sequence A000129 (without the leading 0). - Johannes W. Meijer, Aug 15 2010 Sequence is related to rhombus substitution tilings showing 8-fold rotational symmetry (see A001333). - L. Edson Jeffery, Apr 04 2011 Number of length-n strings of 3 letters {0,1,2} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012 Row sums of A035607, when seen as a triangle read by rows. - Reinhard Zumkeller, Jul 20 2013 REFERENCES A. Froehlich and M. J. Taylor, Algebraic Number Theory, Cambridge, 1991 (see p. 3). Thomas Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8 Tanya Khovanova, Recursive Sequences Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374. Shiva Samieinia, Digital straight line segments and curves, Licentiate Thesis, Stockholm University, Department of Mathematics, Report 2007:6. S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 75-89 Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63). Index entries for linear recurrences with constant coefficients, signature (2,1). FORMULA a(n) = 2*a(n-1) + a(n-2); a(0)=1; a(1)=3. - Wayne VanWeerthuizen, May 02 2004 a(n) = 2*a(n-1) + a(n-2); a(n+1)/a(n) tends to silver ratio 1+sqrt(2) as n tends to infinity. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007 a(n) = Sum_{k, 0<=k<=n}A147720(n,k)*3^k*(-1/3)^(n-k). - Philippe Deléham, Nov 15 2008 a(n) = (1/2)*[1+sqrt(2)]^n-(1/2)*sqrt(2)*[1-sqrt(2)]^n+(1/2)*[1-sqrt(2)]^n+(1/2)*[1+sqrt(2)]^n *sqrt(2), with n>=0. - Paolo P. Lava, Nov 20 2008 a(n) = Pell(n)+Pell(n+1) with Pell(n) = A000129(n). - Johannes W. Meijer, Aug 15 2010 G.f.: G(0)/(2*x) -1/x, where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013 a(n) = T(n+1, i) / i^(n+1) , where T(n, x) denotes the Chebyshev polynomial of the first kind. - Michael Somos, Jul 28 2018 EXAMPLE G.f. = 1 + 3*x + 7*x^2 + 17*x^3 + 41*x^4 + 99*x^5 + 239*x^6 + 577*x^7 + ... - Michael Somos, Jul 28 2018 MATHEMATICA Expand[Table[((1 + Sqrt)^n + (1 - Sqrt)^n)/2, {n, 1, 30}]] (* Artur Jasinski, Dec 10 2006 *) CoefficientList[Series[(1 + x)/(1 - 2 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2014 *) a[ n_] := ChebyshevT[n+1, I] / I^(n+1); (* Michael Somos, Jul 28 2018 *) PROG (Haskell) a078057 = sum . a035607_row  -- Reinhard Zumkeller, Jul 20 2013 (PARI) {a(n) = polchebyshev(n+1, 1, I) / I^(n+1)}; /* Michael Somos, Jul 28 2018 */ CROSSREFS Essentially the same as A001333, which has many more references. Cf. A131887, A131935, A000129. Sequence in context: A089737 A123335 A001333 * A089742 A187258 A131721 Adjacent sequences:  A078054 A078055 A078056 * A078058 A078059 A078060 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Nov 17 2002 STATUS approved

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Last modified September 27 15:39 EDT 2022. Contains 357062 sequences. (Running on oeis4.)