%I A001353 M3499 N1420
%S A001353 0,1,4,15,56,209,780,2911,10864,40545,151316,564719,2107560,7865521,
%T A001353 29354524,109552575,408855776,1525870529,5694626340,21252634831,
%U A001353 79315912984,296011017105,1104728155436,4122901604639,15386878263120
%N A001353 a(n) = 4a(n-1)-a(n-2) with a(0) = 0, a(1) = 1.
%C A001353 3*a(n)^2 + 1 is a perfect square.
%C A001353 Number of spanning trees in 2 X n grid: by examining what happens at
the right-hand end we see that a(n) = 3*a(n-1) + 2*a(n-2) + 2*a(n-3)
+ ... + 2*a(1) + 1, where the final 1 corresponds to the tree ==...=|
!. Solving this we get a(n) = 4a(n-1) - a(n-2).
%C A001353 Complexity of 2 X n grid.
%C A001353 A016064 also describes triangles whose sides are consecutive integers
and in which an inscribed circle has an integer radius. A001353 is
exactly and precisely mapped to the integer radii of such inscribed
circles, i.e. for each term of A016064, the corresponding term of
A001353 gives the radius of the inscribed circle - Harvey P. Dale
(hpd1(AT)is2.nyu.edu), Dec 28 2000
%C A001353 If M is any term of the sequence, the next one is 2M + sqrt(3M^2 + 1).
- Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 18 2002
%C A001353 n such that 3*n^2=floor(sqrt(3)*n*ceil(sqrt(3)*n)) Benoit Cloitre (benoit7848c(AT)orange.fr),
May 10 2003
%C A001353 For n>0, ratios a(n+1)/a(n) may be obtained as convergents of the continued
fraction expansion of 2+sqrt(3): either as successive convergents
of [4;-4] or as odd convergents of [3;1, 2]. - Lekraj Beedassy (blekraj(AT)yahoo.com),
Sep 19 2003
%C A001353 Ways of packing a 3 X (2n-1) rectangle with dominoes, after attaching
an extra square to the end of one of the sides of length 3. With
reference to A001835, therefore: a(n) = a(n-1) + A001835(n-1) and
A001835(n) = 3*A011835(n-1) + 2*a(n-1). - Joshua Zucker and the Castilleja
School Math Club (joshua_zucker(AT)castilleja.org), Oct 28 2003
%C A001353 a(n+1) is a Chebyshev transform of 4^n, where the sequence with g.f.
G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)). -
Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
%C A001353 This sequence generates many brilliant (A078972) numbers for a(p) with
prime p: a(2) = 4 = 2 * 2 a(3) = 15 = 3 * 5 a(5) = 209 = 11 * 19
a(7) = 2911 = 41 * 71 a(19) = 21252634831 = 110771 * 191861 a(37)
= 419245718107612602961 = 15558008491 * 26947261171. Is this a prime-free
sequence? If not, what is its first prime? - Jonathan Vos Post (jvospost3(AT)gmail.com),
Feb 08 2005
%C A001353 Numbers such that there is an m with t(n+m)=3t(m), where t(n) are the
triangular numbers A000217. For instance t(35)=3t(20)=630, so 35-20=15
is in the sequence. - comment by Floor van Lamoen (fvlamoen(AT)hotmail.com),
Oct 13 2005
%C A001353 a(n) = number of unique matrix products in (A+B+C+D)^n where commutator
[A,B]=0 but neither A nor B commutes with C or D. - Paul D. Hanna
and Max Alekseyev (maxale(AT)gmail.com), Feb 01 2006
%C A001353 For n>1, middle side (or long leg) of primitive Pythagorean triangles
having an angle nearing pi/3 with larger values of sides. [Complete
triple (X, Y, Z), X<Y<Z, is given by X=A120892(n), Y=a(n), Z=A120893(n),
with recurrence relations X(i+1)=2*{X(i) - (-1)^i} + a(i) ; Z(i+1)=2*{Z(i)
+ a(i)} - (-1)^i] - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 13
2006
%C A001353 Values y solving the Pellian x^2 - 3*y^2 = 1; Corresponding x given by
A001075(n). Moreover, we have a(n) = 2*a(n-1) + A001075(n-1). - Lekraj
Beedassy (blekraj(AT)yahoo.com), Jul 13 2006
%C A001353 Number of 2 X n simple rectangular mazes. A simple rectangular m X n
maze is a graph G with vertex set {0,1,...,m} X {0,1,...,n} that
satsifies the following two properties: (i) G consists of two orthogonal
trees; (ii) one tree has a path that sequentially connects (0,0),
(0,1),...,(0,n),(1,n),...,(m-1, n) and the other tree has a path
that sequentially connects (1,0),(2,0),...,(m,0),(m,1),...,(m,n).
For example, a(2)=4 because there are four 2X2 simple rectangular
mazes:
%C A001353 .__.............__ ......__.............__
%C A001353 |..|..|........|__...|.......|.....|........|...__|
%C A001353 |...__|........|...__|.......|..|__|........|...__|. - Dennis P. Walsh
(dwalsh(AT)mtsu.edu), Oct 04 2006
%C A001353 [1,4,15,56,209,...] is the Hankel transform of [1,1,5,26,139,758,...](see
A005573). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 14 2007
%C A001353 The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4,
26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075,
denominators=A001353. - Clark Kimberling (ck6(AT)evansville.edu),
Aug 27 2008
%C A001353 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009:
(Start)
%C A001353 A001353 and A001835 = bisection of continued fraction [1,2,1,2,1,2,...]
%C A001353 i.e. of [1, 3, 4, 11, 15, 41,...].
%C A001353 a(n) = determinant of an n*n tridiagonal matrix with one's in the super
and
%C A001353 subdiagonals and (4,4,4,...) as the main diagonal.
%C A001353 A001835 and A001353 = right and next to right borders of triangle A125077
(End)
%C A001353 Number of units of a(n) belongs to a periodic sequence: 0, 1, 4, 5, 6,
9. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
%D A001353 M. N. Deshpande, One Interesting Family of Diophantine Triplets, International
Journal of Mathematical Education In Science and Technology, Vol.
33 (No. 2, Mar-Apr), 2002.
%D A001353 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence
Sequences, Amer. Math. Soc., 2003; p. 163.
%D A001353 E. I. Emerson, Recurrent sequences in the equation DQ^2 = R^2 + N, Fib.
Quart., 7 (1969), 231-242.
%D A001353 F. Faase, On the number of specific spanning subgraphs of the graphs
G X P_n, Ars Combin. 49 (1998), 129-154.
%D A001353 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley,
Reading, MA, 1990, p. 329.
%D A001353 T. N. E. Greville, Table for third-degree spline interpolations with
equally spaced arguments, Math. Comp., 24 (1970), 179-183.
%D A001353 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib.
Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=4, q=-1.
%D A001353 W. D. Hoskins, Table for third-degree spline interpolation using equi-spaced
knots, Math. Comp., 25 (1971), 797-801.
%D A001353 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p.
104.
%D A001353 G. Kreweras, Complexite et circuits Euleriens dans la sommes tensorielles
de graphes, J. Combin. Theory, B 24 (1978), 202-212.
%D A001353 W. Lang, On polynomials related to powers of the generating function
of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq. (44) lhs,
m=6.
%D A001353 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag.,
40 (1967), 74-83.
%D A001353 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
" Elemente der Mathematik, 52 (1997) 122-126.
%D A001353 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley,
New York, 1966.
%D A001353 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001353 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A001353 T. D. Noe, <a href="b001353.txt">Table of n, a(n) for n=0..200</a>
%H A001353 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A001353 F. Faase, <a href="http://www.iwriteiam.nl/Cpaper.zip">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 A001353 F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamilton
cycles in product graphs</a>
%H A001353 F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from
the counting program</a>
%H A001353 F. Faase, <a href="http://home.wxs.nl/~faase009/counting.html">Counting
Hamilton cycles in product graphs</a>
%H A001353 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A001353 Hojoo Lee, <a href="http://www.math.uu.nl/people/beukers/getaltheorie/
pen0795.pdf">Problems in elementary number theory</a> Problem I 18.
%H A001353 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A001353 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001353 P. Raff <a href="http://www.math.rutgers.edu/~praff/span/2/12/index.xml">
Analysis of the Number of Spanning Trees of K_2 x P_n</a>. Contains
sequence, recurrence, generating function, and more. [From Paul Raff
(praff(AT)math.rutgers.edu), Mar 06 2009]
%H A001353 D. P. Walsh, <a href="http://www.mtsu.edu/~dwalsh/MAZECNT2.pdf">Counting
n x 2 Simple Rectangular Mazes</a>
%H A001353 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%F A001353 a(n)=[(2+sqrt(3))^n-(2-sqrt(3))^n]/(2*sqrt(3)).
%F A001353 Limit as n-> infinity of a(n)/a(n-1) = 2 + sqrt(3). - Gregory V. Richardson
(omomom(AT)hotmail.com), Oct 06 2002
%F A001353 Binomial transform of A002605. E.g.f.: exp(2x)sinh(sqrt(3)x)/sqrt(3).
%F A001353 G.f.: x/(1-4*x+x^2). a(n) = S(n-1, 4) = U(n-1, 2), S(-1, x) := 0, Chebyshev's
polynomials of the second kind A049310.
%F A001353 a(n+1)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*4^(n-2k)} - Paul Barry
(pbarry(AT)wit.ie), Oct 25 2004
%F A001353 a(n)=sum{k=0..n-1, binomial(n+k, 2k+1)2^k} - Paul Barry (pbarry(AT)wit.ie),
Nov 30 2004
%F A001353 a(n)=3*a(n-1)+3*a(n-2)-a(n-3); a(0)=0, a(1)=1, a(2)=4. - Lekraj Beedassy
(blekraj(AT)yahoo.com), Jul 13 2006
%F A001353 a(n) = 2*a(n-1)+sqrt[3*a(n-1)^2+1]. a(n) = -A106707(n). - R. J. Mathar
(mathar(AT)strw.leidenuniv.nl), Jul 07 2006
%F A001353 a(n) = 3*(a(n-1)+a(n-2))-a(n-3), a(n) = 5*(a(n-1)-a(n-2))+a(n-3). - Mohamed
Bouhamida (bhmd95(AT)yahoo.fr), Sep 20 2006
%F A001353 M^n * [1,0] = [A001075(n), A001353(n)], where M = the 2 X 2 matrix [2,
3; 1,2]; e.g., a(4) = 56 since M^4 * [1,0] = [97, 56] = [A001075(4),
A001353(4)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2006
%F A001353 Sequence satisfies 1 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v.
- Michael Somos Sep 19 2008
%F A001353 a(-n) = -a(n). - Michael Somos Sep 19 2008
%e A001353 For example, when n=3:
%e A001353 ****
%e A001353 .***
%e A001353 .***
%e A001353 can be packed with dominoes in 4 different ways: 3 in which the top row
is tiled with two horizontal dominoes and 1 in which the top row
has two vertical and one horizontal domino, as shown below, so a(2)
= 4.
%e A001353 ---- ---- ---- ||--
%e A001353 .||| .--| .|-- .|||
%e A001353 .||| .--| .|-- .|||
%p A001353 A001353 := proc(n) option remember; if n <= 1 then 1+3*n else 4*A001353(n-1)-A001353(n-2);
fi; end;
%p A001353 A001353:=z/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.]
%t A001353 a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[
a[n], {n, 0, 23}]] (from Robert G. Wilson v Jan 13 2005)
%t A001353 lst = {}; Do[AppendTo[lst, GegenbauerC[n, 1, 2]], {n, -1, 23}]; lst [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 14 2009]
%o A001353 (PARI) M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=0,30,print1(([1,0,0]*M^i)[2],
",")) - from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005
%o A001353 sage: [lucas_number1(n,4,1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 25 2008
%o A001353 (PARI) {a(n) = real( (2 + quadgen(12))^n / quadgen(12) )} /* Michael
Somos Sep 19 2008 */
%o A001353 (PARI) {a(n) = subst( polchebyshev(n-1, 2), x, 2)} /* Michael Somos Sep
19 2008 */
%o A001353 (Other) sage: [lucas_number1(n,4,1) for n in xrange(0, 25)]# [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%Y A001353 a(n) = sqrt((A001075(n)^2-1)/3).
%Y A001353 Cf. A003500, A001835, A001075, A001571, A001834, A002531, A005246, A016064,
A082840, A079935, A078972.
%Y A001353 A bisection of A002530.
%Y A001353 Cf. A125077 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009]
%Y A001353 Sequence in context: A060111 A077824 A010905 this_sequence A106707 A125905
A026030
%Y A001353 Adjacent sequences: A001350 A001351 A001352 this_sequence A001354 A001355
A001356
%K A001353 nonn,easy,nice
%O A001353 0,3
%A A001353 N. J. A. Sloane (njas(AT)research.att.com).
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