Search: id:A001835 Results 1-1 of 1 results found. %I A001835 M2894 N1160 %S A001835 1,1,3,11,41,153,571,2131,7953,29681,110771,413403,1542841,5757961, %T A001835 21489003,80198051,299303201,1117014753,4168755811,15558008491, %U A001835 58063278153,216695104121,808717138331,3018173449203,11263976658481 %N A001835 a(n) = 4a(n-1) - a(n-2); a(0)=a(1)=1. %C A001835 See A079935 for another version. %C A001835 Number of ways of packing a 3 X 2(n-1) rectangle with dominoes. - David Singmaster. %C A001835 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 %C A001835 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 A001835 Terms are the solutions to: 3x^2-2 is a square. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002 %C A001835 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 %C A001835 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 %C A001835 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 %C A001835 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 %C A001835 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 %C A001835 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 %C A001835 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009: (Start) %C A001835 A001835 and A001353 = bisection of denominators of continued fraction %C A001835 [1, 2, 1, 2, 1, 2,...]; i.e. bisection of [1, 3, 4, 11, 15, 41, 56,...]. %C A001835 A001835 and A001353 = rightmost border and adjacent diagonal of triangle A125077. %C A001835 a(n) = determinant of an n*n tridiagonal matrix with 1's in the super and %C A001835 subdiagonals and (3,4,4,4,...) as the main diagonal. Also, the product of %C A001835 the eigenvalues of such matrices and a(n) = PRODUCT_(k=1..(n-1)/2)} (4 + 2*Cos 2kPi/n. %C A001835 (End) %D A001835 L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375. %D A001835 F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154. %D A001835 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329. %D A001835 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). %D A001835 Clark Kimberling, "Best lower and upper approximates to irrational numbers, " Elemente der Mathematik, 52 (1997) 122-126. %D A001835 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. %D A001835 Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640. %D A001835 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A001835 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A001835 R. P. Stanley, Enumerative Combinatorics I, p. 292. %D A001835 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83. %H A001835 T. D. Noe, Table of n, a(n) for n=0..200 %H A001835 L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil. %H A001835 F. Faase, Counting Hamilton cycles in product graphs %H A001835 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. %H A001835 F. Faase, Counting Hamilton cycles in product graphs %H A001835 F. Faase, Results from the counting program %H A001835 Tanya Khovanova, Recursive Sequences %H A001835 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 409 %H A001835 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. %H A001835 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001835 Index entries for sequences related to dominoes %H A001835 Index entries for sequences related to Chebyshev polynomials. %H A001835 Index entries for sequences related to linear recurrences with constant coefficients %F A001835 a(n) = ((3+sqrt(3))^(2*n-1)+(3-sqrt(3))^(2*n-1))/6^n. - Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 01 2002 %F A001835 a(n)=(8+a(n-1)a(n-2))/a(n-3) - Michael Somos, Aug 01, 2001 %F A001835 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 %F A001835 Lim. n -> Inf. a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002 %F A001835 a(n)=2*A061278(n-1)+1 for n>0 - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002 %F A001835 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 %F A001835 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. %F A001835 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 A001835 a(n+1)=sqrt((A001834(n)^2 + 2)/3), n>=0 (see Cloitre comment). %F A001835 G.f.: (1-3*x)/(1-4*x+x^2). a(1-n)=a(n). %F A001835 a(1-n)=a(n). - Michael Somos Aug 07 2006 %F A001835 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 A001835 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] %p A001835 A001835:=-(-1+3*z)/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.] %p A001835 f:=n->((3+sqrt(3))^(2*n-1)+(3-sqrt(3))^(2*n-1))/6^n; [seq(simplify(expand(f(n))), n=0..20)]; [njas, Nov 10 2009] %o A001835 (PARI) {a(n) = real( (2 + quadgen(12))^n * (1 - 1 / quadgen(12)) )} / * Michael Somos Sep 19 2008 */ %o A001835 (PARI) {a(n) = subst( (polchebyshev(n) + polchebyshev(n-1)) / 3, x, 2)} /* Michael Somos Sep 19 2008 */ %o A001835 (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] %Y A001835 Cf. A001519 (similar summation) %Y A001835 Row 3 of array A099390. %Y A001835 Essentially the same as A079935. %Y A001835 First differences of A001353. Partial sums of A052530. Pairwise sums of A006253. Bisection of A002530, A005246 and A048788. Cf. A003699, A082841. %Y A001835 First column of array A103997. %Y A001835 Cf. A101265. %Y A001835 Cf. A125077 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009] %Y A001835 Sequence in context: A086972 A077831 A032952 this_sequence A079935 A113437 A076540 %Y A001835 Adjacent sequences: A001832 A001833 A001834 this_sequence A001836 A001837 A001838 %K A001835 nonn,easy,nice,new %O A001835 0,3 %A A001835 N. J. A. Sloane (njas(AT)research.att.com). %E A001835 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002 Search completed in 0.002 seconds