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A001835 a(n) = 4*a(n-1) - a(n-2), with a(0)=1, a(1)=1.
(Formerly M2894 N1160)
60
1, 1, 3, 11, 41, 153, 571, 2131, 7953, 29681, 110771, 413403, 1542841, 5757961, 21489003, 80198051, 299303201, 1117014753, 4168755811, 15558008491, 58063278153, 216695104121, 808717138331, 3018173449203, 11263976658481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

See A079935 for another version.

Number of ways of packing a 3 X 2(n-1) rectangle with dominoes. - David Singmaster.

Equivalently, number of perfect matchings of the P_3 X P_{2(n-1)} lattice graph. - Emeric Deutsch, Dec 28 2004

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

Terms are the solutions to: 3x^2-2 is a square. - Benoit Cloitre, Apr 07 2002

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

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

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

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

sqrt(3) = 2/2 + 2/3 + 2/(3*11) + 2/(11*41) + 2/(41*153) + 2/(153*571)... - Gary W. Adamson, Dec 18 2007

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

From Gary W. Adamson, Jun 21 2009: (Start)

A001835 and A001353 = bisection of denominators of continued fraction [1, 2, 1, 2, 1, 2,...]; i.e. bisection of A002530.

a(n) = determinant of an n*n tridiagonal matrix with 1's in the super- and sub-diagonals and (3,4,4,4,...) as the main diagonal.

Also, the product of the eigenvalues of such matrices: a(n) = PRODUCT_{k=1..(n-1)/2)} (4 + 2*Cos 2kPi/n).

(End)

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->inf} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 27 2010

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]

For n>= 2, a(n) equals the permanent of the (2n-2)X(2n-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]

Primes in the sequence are apparently those in A096147. - R. J. Mathar, May 09 2013

Except for the first term, positive values of x (or y) satisfying x^2 - 4xy + y^2 + 2 = 0. - Colin Barker, Feb 04 2014

Except for the first term, positive values of x (or y) satisfying x^2 - 14xy + y^2 + 32 = 0. - Colin Barker, Feb 10 2014

The (1,1) element of A^n where A=(1,1,1;1,2,1;1,1,2). - David Neil McGrath, Jul 23 2014

REFERENCES

Bastida, Julio R. 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)

J. B. Cosgrave and K. Dilcher, A role for generalized Fermat numbers, Math. Comp., to appear 2016; http://johnbcosgrave.com/publications.php (See paper #10).

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375.

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

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).

R. P. Stanley, Enumerative Combinatorics I, p. 292.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Steve Butler, Paul Horn, and Eric Tressler, Intersecting Domino Tilings, Fibonacci Quart. 48 (2010), no. 2, 114-120.

A. Consilvio et al., Tilings, ordered partitions, and weird languages, MAA FOCUS, June/July 2012, 16-17.

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.

F. Faase, Counting Hamilton cycles in product graphs

F. Faase, Results from the counting program

H. Hosoya and A. Motoyama, 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, J. Math. Physics 26 (1985) 157-167 (Table V).

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 409

Tanya Khovanova, Recursive Sequences

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

David Klarner, Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52.

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], 2013, Table 2.

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.

F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.

Index entries for sequences related to dominoes

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (4,-1).

FORMULA

G.f.: (1-3*x)/(1-4*x+x^2). -  Simon Plouffe in his 1992 dissertation

a(1-n) = a(n).

a(n) = ((3+sqrt(3))^(2*n-1)+(3-sqrt(3))^(2*n-1))/6^n. - Dean Hickerson, Dec 01 2002

a(n) = (8+a(n-1)*a(n-2))/a(n-3). - Michael Somos, Aug 01, 2001

a(n+1) = sum(2^k * binomial(n+k, n-k), k=0..n), n>=0. - Len Smiley, Dec 09 2001

Lim. n -> Inf. a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson, Oct 10 2002

a(n) = 2*A061278(n-1)+1 for n>0. - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002

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

a(n+1)= sum(((-1)^k)*((2*n+1)/(2*n+1-k))*binomial(2*n+1-k,k)*6^(n-k),k=0..n) (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

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.

a(n+1) = sqrt((A001834(n)^2 + 2)/3), n>=0 (see Cloitre comment).

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

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). [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]

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]

a(n) = (a(n-1)^2 + 2)/a(n-2). - Irene Sermon, Oct 28 2013

a(n) = A001353(n+1) -3*A001353(n). - R. J. Mathar, Oct 30 2015

MAPLE

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

MATHEMATICA

CoefficientList[Series[(1-3x)/(1-4x+x^2), {x, 0, 24}], x] (* Jean-François Alcover, Jul 25 2011, after g.f. *)

LinearRecurrence[{4, -1}, {1, 1}, 30] (* Harvey P. Dale, Jun 08 2013 *)

PROG

(PARI) {a(n) = real( (2 + quadgen(12))^n * (1 - 1 / quadgen(12)) )} /* Michael Somos, Sep 19 2008 */

(PARI) {a(n) = subst( (polchebyshev(n) + polchebyshev(n-1)) / 3, x, 2)} /* Michael Somos, Sep 19 2008 */

(Sage) [lucas_number1(n, 4, 1)-lucas_number1(n-1, 4, 1) for n in xrange(0, 25)] # Zerinvary Lajos, Apr 29 2009

(Haskell)

a001835 n = a001835_list !! n

a001835_list =

   1 : 1 : zipWith (-) (map (4 *) $ tail a001835_list) a001835_list

-- Reinhard Zumkeller, Aug 14 2011

CROSSREFS

Row 3 of array A099390.

Essentially the same as A079935.

First differences of A001353.

Partial sums of A052530.

Pairwise sums of A006253.

Bisection of A002530, A005246 and A048788.

First column of array A103997.

Cf. A001519, A003699, A082841, A101265, A125077, A001353, A001542.

Sequence in context: A077831 A032952 * A079935 A113437 A076540 A196472

Adjacent sequences:  A001832 A001833 A001834 * A001836 A001837 A001838

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 27 20:41 EDT 2016. Contains 275912 sequences.