0,3

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 (deutsch(AT)duke.poly.edu), 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 (benoit7848c(AT)orange.fr), Apr 07 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.

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 (benoit7848c(AT)orange.fr), 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 (reinhard.zumkeller(AT)gmail.com), 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 (blekraj(AT)yahoo.com), 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. n=2, we have 02, 03, 11, 12, 13, 21, 22, 23, 31, 32, 33) - Tanya Khovanova (tanyakh(AT)yahoo.com), Jan 10 2007

Sqrt(3) = 2/2 + 2/3 + 2/(3*11) + 2/(11*41) + 2/(41*153) + 2/(153*571)... - Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 (ck6(AT)evansville.edu), Aug 27 2008

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009: (Start)

A001835 and A001353 = bisection of denominators of continued fraction

[1, 2, 1, 2, 1, 2,...]; i.e. bisection of [1, 3, 4, 11, 15, 41, 56,...].

A001835 and A001353 = rightmost border and adjacent diagonal of triangle A125077.

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. Also, the product of

the eigenvalues of such matrices and a(n) = PRODUCT_(k=1..(n-1)/2)} (4 + 2*Cos 2kPi/n.

(End)

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

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

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

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).

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

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

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

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

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

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

Index entries for sequences related to linear recurrences with constant coefficients

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

Tanya Khovanova, Recursive Sequences

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

F. Faase, Counting Hamilton cycles in product graphs

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 409

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Index entries for sequences related to dominoes

Index entries for sequences related to Chebyshev polynomials.

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 (smiley(AT)math.uaa.alaska.edu), Dec 09 2001

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

a(n) = ((3+sqrt(3))^(2n-1)+(3-sqrt(3))^(2n-1))/6^n. - Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 01 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 (benoit7848c(AT)orange.fr), Nov 10 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).

G.f.: (1-3*x)/(1-4*x+x^2). a(1-n)=a(n).

a(1-n)=a(n). - Michael Somos Aug 07 2006

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

A001835:=-(-1+3*z)/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.]

(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 */

(Other) sage: [lucas_number1(n, 4, 1)-lucas_number1(n-1, 4, 1) for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2009]

Cf. A001519 (similar summation)

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. Cf. A003699, A082841.

First column of array A103997.

Cf. A101265.

Adjacent sequences: A001832 A001833 A001834 this_sequence A001836 A001837 A001838

A125077 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009]

Sequence in context: A086972 A077831 A032952 this_sequence A079935 A113437 A076540

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002