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%I M2894 N1160 #280 Mar 31 2024 17:25:45
%S 1,1,3,11,41,153,571,2131,7953,29681,110771,413403,1542841,5757961,
%T 21489003,80198051,299303201,1117014753,4168755811,15558008491,
%U 58063278153,216695104121,808717138331,3018173449203,11263976658481
%N a(n) = 4*a(n-1) - a(n-2), with a(0) = 1, a(1) = 1.
%C See A079935 for another version.
%C Number of ways of packing a 3 X 2*(n-1) rectangle with dominoes. - David Singmaster.
%C Equivalently, number of perfect matchings of the P_3 X P_{2(n-1)} lattice graph. - _Emeric Deutsch_, Dec 28 2004
%C The terms of this sequence are the positive square roots of the indices of the octagonal numbers (A046184) - Nicholas S. Horne (nairon(AT)loa.com), Dec 13 1999
%C Terms are the solutions to: 3*x^2 - 2 is a square. - _Benoit Cloitre_, Apr 07 2002
%C Gives solutions x > 0 of the equation floor(x*r*floor(x/r)) == floor(x/r*floor(x*r)) where r = 1 + sqrt(3). - _Benoit Cloitre_, Feb 19 2004
%C a(n) = L(n-1,4), where L is defined as in A108299; see also A001834 for L(n,-4). - _Reinhard Zumkeller_, Jun 01 2005
%C Values x + y, where (x, y) solves for x^2 - 3*y^2 = 1, i.e., a(n) = A001075(n) + A001353(n). - _Lekraj Beedassy_, Jul 21 2006
%C Number of 01-avoiding words of length n on alphabet {0,1,2,3} which do not end in 0. (E.g., for n = 2 we have 02, 03, 11, 12, 13, 21, 22, 23, 31, 32, 33.) - _Tanya Khovanova_, Jan 10 2007
%C sqrt(3) = 2/2 + 2/3 + 2/(3*11) + 2/(11*41) + 2/(41*153) + 2/(153*571) + ... - _Gary W. Adamson_, Dec 18 2007
%C The lower principal convergents to 3^(1/2), beginning with 1/1, 5/3, 19/11, 71/41, comprise a strictly increasing sequence; numerators = A001834, denominators = A001835. - _Clark Kimberling_, Aug 27 2008
%C From _Gary W. Adamson_, Jun 21 2009: (Start)
%C A001835 and A001353 = bisection of denominators of continued fraction [1, 2, 1, 2, 1, 2, ...]; i.e., bisection of A002530.
%C a(n) = determinant of an n*n tridiagonal matrix with 1's in the super- and subdiagonals and (3, 4, 4, 4, ...) as the main diagonal.
%C Also, the product of the eigenvalues of such matrices: a(n) = Product_{k=1..(n-1)/2)} (4 + 2*cos(2*k*Pi/n).
%C (End)
%C Let M = a triangle with the even-indexed Fibonacci numbers (1, 3, 8, 21, ...) in every column, and the leftmost column shifted up one row. a(n) starting (1, 3, 11, ...) = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. - _Gary W. Adamson_, Jul 27 2010
%C a(n+1) is the number of compositions of n when there are 3 types of 1 and 2 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%C For n >= 2, a(n) equals the permanent of the (2*n-2) X (2*n-2) tridiagonal matrix with sqrt(2)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 08 2011
%C Primes in the sequence are apparently those in A096147. - _R. J. Mathar_, May 09 2013
%C Except for the first term, positive values of x (or y) satisfying x^2 - 4xy + y^2 + 2 = 0. - _Colin Barker_, Feb 04 2014
%C Except for the first term, positive values of x (or y) satisfying x^2 - 14xy + y^2 + 32 = 0. - _Colin Barker_, Feb 10 2014
%C The (1,1) element of A^n where A = (1, 1, 1; 1, 2, 1; 1, 1, 2). - _David Neil McGrath_, Jul 23 2014
%C Yong Hao Ng has shown that for any n, a(n) is coprime with any member of A001834 and with any member of A001075. - _René Gy_, Feb 25 2018
%C a(n+1) is the number of spanning trees of the graph T_n, where T_n is a 2 X n grid with an additional vertex v adjacent to (1,1) and (2,1). - _Kevin Long_, May 04 2018
%C a(n)/A001353(n) is the resistance of an n-ladder graph whose edges are replaced by one-ohm resistors. The resistance in ohms is measured at two nodes at one end of the ladder. It approaches sqrt(3) - 1 for n -> infinity. See A342568, A357113, and A357115 for related information. - _Hugo Pfoertner_, Sep 17 2022
%C a(n) is the number of ways to tile a 1 X (n-1) strip with three types of tiles: small isosceles right triangles (with small side length 1), 1 X 1 squares formed by joining two of those right triangles along the hypotenuse, and large isosceles right triangles (with large side length 2) formed by joining two of those right triangles along a short leg. As an example, here is one of the a(6)=571 ways to tile a 1 X 5 strip with these kinds of tiles:
%C ______________
%C | / \ |\ /| |
%C |/___\|_\_/_|__|. - _Greg Dresden_ and Arjun Datta, Jun 30 2023
%D R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
%D Julio R. Bastida, Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009)
%D L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375.
%D F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.
%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics I, p. 292.
%H T. D. Noe, <a href="/A001835/b001835.txt">Table of n, a(n) for n = 0..200</a>
%H Mudit Aggarwal and Samrith Ram, <a href="https://arxiv.org/abs/2206.04437">Generating functions for straight polyomino tilings of narrow rectangles</a>, arXiv:2206.04437 [math.CO], 2022.
%H Krassimir T. Atanassov and Anthony G. Shannon, <a href="https://doi.org/10.7546/nntdm.2020.26.3.218-223">On intercalated Fibonacci sequences</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 218-223.
%H Steve Butler, Paul Horn, and Eric Tressler, <a href="http://www.fq.math.ca/Papers1/48-2/Butler_Horn_Tressler.pdf">Intersecting Domino Tilings</a>, Fibonacci Quart. 48 (2010), no. 2, 114-120.
%H Niccolò Castronuovo, <a href="https://arxiv.org/abs/2102.02739">On the number of fixed points of the map gamma</a>, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
%H A. Consilvio et al., <a href="http://www.maa.org/pubs/FOCUSJun-Jul12_tanton.html">Tilings, ordered partitions, and weird languages</a>, MAA FOCUS, June/July 2012, 16-17.
%H J. B. Cosgrave and K. Dilcher, <a href="http://johnbcosgrave.com/publications.php">A role for generalized Fermat numbers</a>, Math. Comp., to appear 2016; (See paper #10).
%H J. B. Cosgrave and K. Dilcher, <a href="https://doi.org/10.1090/mcom/3111">A role for generalized Fermat numbers</a>, Math. Comp. 86 (2017), 899-933.
%H L. Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</a>.
%H F. Faase, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.2790">On the number of specific spanning subgraphs of the graphs G X P_n</a>, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>
%H Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H Darren B. Glass, <a href="https://doi.org/10.1016/j.ejc.2016.09.010">Critical groups of graphs with dihedral actions. II</a>, Eur. J. Comb. 61, 25-46 (2017).
%H H. Hosoya and A. Motoyama, <a href="http://dx.doi.org/10.1063/1.526778">An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices</a>, J. Math. Physics 26 (1985) 157-167 (Table V).
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=409">Encyclopedia of Combinatorial Structures 409</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997) 122-126.
%H David Klarner and Jordan Pollack, <a href="http://dx.doi.org/10.1016/0012-365X(80)90098-9">Domino tilings of rectangles with fixed width</a>, Disc. Math. 32 (1980) 45-52.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1311.6135">Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings</a>, arXiv:1311.6135 [math.CO], 2013, Table 2.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1406.7788">Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices</a>, arXiv:1406.7788 (2014), eq. (4).
%H Valcho Milchev and Tsvetelina Karamfilova, <a href="https://arxiv.org/abs/1707.09741">Domino tiling in grid - new dependence</a>, arXiv:1707.09741 [math.HO], 2017.
%H Yong Hao Ng, <a href="https://math.stackexchange.com/a/2664328/130022">A partition in three classes of the set of all prime numbers?</a>, Math StackExchange.
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Jaime Rangel-Mondragon, <a href="https://web.archive.org/web/20190411024906/http://www.mathematica-journal.com/issue/v9i3/polyominoes.html">Polyominoes and Related Families</a>, The Mathematica Journal, 9:3 (2005), 609-640.
%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.
%H David Singmaster, <a href="/A005178/a005178.pdf">Letter to N. J. A. Sloane</a>, Oct 3 1982.
%H Anitha Srinivasan, <a href="https://www.fq.math.ca/Papers1/58-5/srinivasan.pdf">The Markoff-Fibonacci Numbers</a>, Fibonacci Quart. 58 (2020), no. 5, 222-228.
%H Thotsaporn ”Aek” Thanatipanonda, <a href="https://www.fq.math.ca/Papers1/57-5/thanatipanonda.pdf">Statistics of Domino Tilings on a Rectangular Board</a>, Fibonacci Quart. 57 (2019), no. 5, 145-153. See p. 151.
%H Herman Tulleken, <a href="https://www.researchgate.net/publication/333296614_Polyominoes">Polyominoes 2.2: How they fit together</a>, (2019).
%H F. V. Waugh and M. W. Maxfield, <a href="http://www.jstor.org/stable/2688511">Side-and-diagonal numbers</a>, Math. Mag., 40 (1967), 74-83.
%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F G.f.: (1 - 3*x)/(1 - 4*x + x^2). - _Simon Plouffe_ in his 1992 dissertation
%F a(1-n) = a(n).
%F a(n) = ((3 + sqrt(3))^(2*n - 1) + (3 - sqrt(3))^(2*n - 1))/6^n. - _Dean Hickerson_, Dec 01 2002
%F a(n) = (8 + a(n-1)*a(n-2))/a(n-3). - _Michael Somos_, Aug 01 2001
%F a(n+1) = Sum_{k=0..n} 2^k * binomial(n + k, n - k), n >= 0. - _Len Smiley_, Dec 09 2001
%F Limit_{n->oo} a(n)/a(n-1) = 2 + sqrt(3). - _Gregory V. Richardson_, Oct 10 2002
%F a(n) = 2*A061278(n-1) + 1 for n > 0. - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002
%F Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n - i, i); then q(n, 2) = a(n+1). - _Benoit Cloitre_, Nov 10 2002
%F a(n+1) = Sum_{k=0..n} ((-1)^k)*((2*n+1)/(2*n + 1 - k))*binomial(2*n + 1 - k, k)*6^(n - k) (from standard T(n,x)/x, n >= 1, Chebyshev sum formula). The Smiley and Cloitre sum representation is that of the S(2*n, i*sqrt(2))*(-1)^n Chebyshev polynomial. - _Wolfdieter Lang_, Nov 29 2002
%F a(n) = S(n-1, 4) - S(n-2, 4) = T(2*n-1, sqrt(3/2))/sqrt(3/2) = S(2*(n-1), i*sqrt(2))*(-1)^(n - 1), with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x) = 0, S(-2, x) = -1, S(n, 4) = A001353(n+1), T(-1, x) = x.
%F a(n+1) = sqrt((A001834(n)^2 + 2)/3), n >= 0 (see Cloitre comment).
%F Sequence satisfies -2 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v. - _Michael Somos_, Sep 19 2008
%F a(n) = (1/6)*(3*(2 - sqrt(3))^n + sqrt(3)*(2 - sqrt(3))^n + 3*(2 + sqrt(3))^n - sqrt(3)*(2 + sqrt(3))^n) (Mathematica's solution to the recurrence relation). - _Sarah-Marie Belcastro_, Jul 04 2009
%F If p[1] = 3, p[i] = 2, (i > 1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n+1) = det A. - _Milan Janjic_, Apr 29 2010
%F a(n) = (a(n-1)^2 + 2)/a(n-2). - _Irene Sermon_, Oct 28 2013
%F a(n) = A001353(n+1) - 3*A001353(n). - _R. J. Mathar_, Oct 30 2015
%F a(n) = a(n-1) + 2*A001353(n-1). - _Kevin Long_, May 04 2018
%F From _Franck Maminirina Ramaharo_, Nov 11 2018: (Start)
%F a(n) = (-1)^n*(A125905(n) + 3*A125905(n-1)), n > 0.
%F E.g.f.: exp^(2*x)*(3*cosh(sqrt(3)*x) - sqrt(3)*sinh(sqrt(3)*x))/3. (End)
%F From _Peter Bala_, Feb 12 2024: (Start)
%F For n in Z, a(n) = A001353(n) + A001353(1-n).
%F For n, j, k in Z, a(n)*a(n+j+k) - a(n+j)*a(n+k) = 2*A001353(j)*A001353(k). The case j = 1, k = 2 is given above. (End)
%p f:=n->((3+sqrt(3))^(2*n-1)+(3-sqrt(3))^(2*n-1))/6^n; [seq(simplify(expand(f(n))),n=0..20)]; # _N. J. A. Sloane_, Nov 10 2009
%t CoefficientList[Series[(1-3x)/(1-4x+x^2), {x, 0, 24}], x] (* _Jean-François Alcover_, Jul 25 2011, after g.f. *)
%t LinearRecurrence[{4,-1},{1,1},30] (* _Harvey P. Dale_, Jun 08 2013 *)
%t Table[Round@Fibonacci[2n-1, Sqrt[2]], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 15 2016 *)
%t Table[(3*ChebyshevT[n, 2] - ChebyshevU[n, 2])/2, {n, 0, 20}] (* _G. C. Greubel_, Dec 23 2019 *)
%o (PARI) {a(n) = real( (2 + quadgen(12))^n * (1 - 1 / quadgen(12)) )} /* _Michael Somos_, Sep 19 2008 */
%o (PARI) {a(n) = subst( (polchebyshev(n) + polchebyshev(n-1)) / 3, x, 2)} /* _Michael Somos_, Sep 19 2008 */
%o (Sage) [lucas_number1(n,4,1)-lucas_number1(n-1,4,1) for n in range(25)] # _Zerinvary Lajos_, Apr 29 2009
%o (Sage) [(3*chebyshev_T(n,2) - chebyshev_U(n,2))/2 for n in (0..20)] # _G. C. Greubel_, Dec 23 2019
%o (Haskell)
%o a001835 n = a001835_list !! n
%o a001835_list =
%o 1 : 1 : zipWith (-) (map (4 *) $ tail a001835_list) a001835_list
%o -- _Reinhard Zumkeller_, Aug 14 2011
%o (Magma) [n le 2 select 1 else 4*Self(n-1)-Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Sep 16 2016
%o (GAP) a:=[1,1];; for n in [3..20] do a[n]:=4*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 23 2019
%Y Row 3 of array A099390.
%Y Essentially the same as A079935.
%Y First differences of A001353.
%Y Partial sums of A052530.
%Y Pairwise sums of A006253.
%Y Bisection of A002530, A005246 and A048788.
%Y First column of array A103997.
%Y Cf. A001519, A003699, A082841, A101265, A125077, A001353, A001542, A096147 (subsequence of primes).
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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