cp's OEIS Frontend

A078057 Expansion of (1+x)/(1-2*x-x^2).

%I A078057 #106 Aug 18 2025 00:32:25
%S A078057 1,3,7,17,41,99,239,577,1393,3363,8119,19601,47321,114243,275807,
%T A078057 665857,1607521,3880899,9369319,22619537,54608393,131836323,318281039,
%U A078057 768398401,1855077841,4478554083,10812186007,26102926097,63018038201,152139002499,367296043199,886731088897
%N A078057 Expansion of (1+x)/(1-2*x-x^2).
%C A078057 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) ].
%C A078057 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
%C A078057 The number of Khalimsky-continuous functions with one fixed endpoint. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
%C A078057 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
%C A078057 An elephant sequence, see A175655. For the central square six A[5] 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
%C A078057 Sequence is related to rhombus substitution tilings showing 8-fold rotational symmetry (see A001333). - _L. Edson Jeffery_, Apr 04 2011
%C A078057 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
%C A078057 Row sums of A035607, when seen as a triangle read by rows. - _Reinhard Zumkeller_, Jul 20 2013
%C A078057 Interpretation via Cartier-Foata traces. Consider the trace monoid on Sigma = {a, b, c} in which only a and b commute. Let h(t) be the CF height (number of cliques) and |t| the length (number of letters). Then a(n) counts traces t with h(t) = n and |t| = n. In CF normal form these are exactly sequences of n singleton cliques {a},{b},{c} with the rule: whenever two consecutive cliques lie in {{a},{b}} they must be equal (so {a}{b} and {b}{a} are forbidden, while {a}{a}, {b}{b}, and any occurrence of {c} are allowed). Equivalently, no direct change a<->b without an intervening {c}. Examples: valid {a}{a}{c}{b}{b}; invalid {a}{b}{a}. This CF trace model is equivalent to Wayne VanWeerthuizen's bead model given above, where {a} = red, {b} = blue, {c} = green. In both cases, a red bead cannot immediately follow a blue bead, and a blue bead cannot immediately follow a red bead, without a green bead in between. - _Constantinos Kourouzides_, Aug 13 2025
%D A078057 A. Froehlich and M. J. Taylor, Algebraic Number Theory, Cambridge, 1991 (see p. 3).
%D A078057 Thomas Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
%H A078057 Vincenzo Librandi, <a href="/A078057/b078057.txt">Table of n, a(n) for n = 0..1000</a>
%H A078057 Jean-Luc Baril and Nathanaël Hassler, <a href="http://jl.baril.u-bourgogne.fr/interva.pdf">Intervals in a family of Fibonacci lattices</a>, Univ. de Bourgogne (France, 2024). See p. 7.
%H A078057 César Bautista-Ramos and Carlos Guillén-Galván, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Bautista/bautista4.html">Fibonacci numbers of generalized Zykov sums</a>, J. Integer Seq., 15 (2012), Article 12.7.8
%H A078057 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A078057 Constantinos Kourouzides, <a href="https://github.com/Constakour/trace-monoid-paper">Study of an elementary trace monoid with two commuting generators</a>, GitHub repository.
%H A078057 Stephan Mertens, <a href="https://arxiv.org/abs/2408.08053">Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph</a>, arXiv:2408.08053 [math.CO], 2024. See p. 5.
%H A078057 Emanuele Munarini, <a href="http://www.emis.de/journals/INTEGERS/papers/j29/j29.Abstract.html">Combinatorial properties of the antichains of a garland</a>, Integers, 9 (2009), 353-374.
%H A078057 Shiva Samieinia, <a href="http://www2.math.su.se/reports/2007/6/">Digital straight line segments and curves</a>, Licentiate Thesis, Stockholm University, Department of Mathematics, Report 2007:6.
%H A078057 Shiva Samieinia, <a href="http://dx.doi.org/10.4171/PM/1858">The number of continuous curves in digital geometry</a>, Port. Math. 67 (1) (2010) 75-89
%H A078057 Gyula Tasi and Fujio Mizukami, <a href="http://dx.doi.org/10.1023/A:1019163812482">Quantum algebraic-combinatoric study of the conformational properties of n-alkanes</a>, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63).
%H A078057 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%F A078057 a(n) = 2*a(n-1) + a(n-2); a(0)=1; a(1)=3. - _Wayne VanWeerthuizen_, May 02 2004
%F A078057 a(n) = 2*a(n-1) + a(n-2); lim_{n->oo} a(n+1)/a(n) = 1 + sqrt(2) (i.e., the silver ratio). - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
%F A078057 a(n) = Sum_{k=0..n} A147720(n,k)*3^k*(-1/3)^(n-k). - _Philippe Deléham_, Nov 15 2008
%F A078057 a(n) = Pell(n) + Pell(n+1) with Pell(n) = A000129(n). - _Johannes W. Meijer_, Aug 15 2010
%F A078057 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
%F A078057 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
%F A078057 E.g.f.: exp(x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)). - _Stefano Spezia_, Jan 31 2023
%F A078057 a(n) = A000129(n)+A000129(n+1). - _R. J. Mathar_, Mar 19 2025
%e A078057 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
%t A078057 Expand[Table[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2, {n, 1, 30}]] (* _Artur Jasinski_, Dec 10 2006 *)
%t A078057 CoefficientList[Series[(1 + x)/(1 - 2 x - x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 16 2014 *)
%t A078057 a[ n_] := ChebyshevT[n+1, I] / I^(n+1); (* _Michael Somos_, Jul 28 2018 *)
%o A078057 (Haskell)
%o A078057 a078057 = sum . a035607_row  -- _Reinhard Zumkeller_, Jul 20 2013
%o A078057 (PARI) {a(n) = polchebyshev(n+1, 1, I) / I^(n+1)}; /* _Michael Somos_, Jul 28 2018 */
%Y A078057 Essentially the same as A001333, which has many more references.
%Y A078057 Cf. A000129, A001405, A035607, A131887, A131935, A147720, A175655.
%K A078057 nonn,easy,changed
%O A078057 0,2
%A A078057 _N. J. A. Sloane_, Nov 17 2002