%I A001333 M2665 N1064
%S A001333 1,1,3,7,17,41,99,239,577,1393,3363,8119,19601,47321,114243,275807,
%T A001333 665857,1607521,3880899,9369319,22619537,54608393,131836323,318281039,
%U A001333 768398401,1855077841,4478554083,10812186007,26102926097,63018038201
%N A001333 Numerators of continued fraction convergents to sqrt(2).
%C A001333 Number of n-step non-selfintersecting paths starting at (0,0) with steps
of types (1,0), (-1,0) or (0,1) [Stanley].
%C A001333 Number of symmetric 2n X 2 or (2n-1) X 2 crossword puzzle grids: all
white squares are edge connected; at least 1 white square on every
edge of grid; 180 degree rotational symmetry - Erich Friedman (erich.friedman(AT)stetson.edu)
%C A001333 a(n+1) is the number of ways to put molecules on a 2 X n ladder lattice
so that the molecules do not touch each other.
%C A001333 Number of (n-1) X 2 binary arrays with a path of adjacent 1's from top
row to bottom row. - Ron Hardin (rhhardin(AT)att.net), Mar 16 2002
%C A001333 a(2*n+1) with b(2*n+1) := A000129(2*n+1), n>=0, give all (positive integer)
solutions to Pell equation a^2 - 2*b^2 = -1.
%C A001333 a(2*n) with b(2*n) := A000129(2*n), n>=1, give all (positive integer)
solutions to Pell equation a^2 - 2*b^2 = +1 (see Emerson reference).
%C A001333 Bisection: a(2*n)= T(n,3)=A001541(n), n>=0 and a(2*n+1)=S(2*n,2*sqrt(2))=
A002315(n), n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials
of the first,resp. second kind. See A053120, resp. A049310.
%C A001333 Binomial transform of A077957. - Paul Barry (pbarry(AT)wit.ie), Feb 25
2003
%C A001333 For n>0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4
and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 2.
- Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%C A001333 x satisfying x^2 - 2*y^2 = -+1. Corresponding y is given by A000129(n).
- Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 24 2004
%C A001333 For n>1, a(n) corresponds to the longer side of a near right-angled isosceles
triangle, one of the equal sides being A000129(n). - Lekraj Beedassy
(blekraj(AT)yahoo.com), Aug 06 2004
%C A001333 Exponents of terms in the series F(x,1), where F is determined by the
equation F(x,y) = xy + F(x^2*y,x). - Jonathan Sondow (jsondow(AT)alumni.princeton.edu),
Dec 18 2004
%C A001333 Number of n-words from the alphabet A={0,1,2} which two neighbors differ
by at most 1. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk),
Aug 30 2006
%C A001333 Consider the mapping f(a/b) = (a + 2b)/(a + b). Taking a = b = 1 to start
with and carrying out this mapping repeatedly on each new (reduced)
rational number gives the following sequence 1/1, 3/2,7/5,17/12,41/
29,... converging to 2^(1/2). Sequence contains the numerators. -
Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003 [Amended
by Paul E. Black (paul.black(AT)nist.gov), Dec 18 2006]
%C A001333 a(n) mod 10 = A131707. See A131711. - Paul Curtz (bpcrtz(AT)free.fr),
Apr 08 2008
%C A001333 Starting (1, 3, 7, 17,...) = row sums of triangle A140750 - Gary W. Adamson
(qntmpkt(AT)yahoo.com), May 26 2008
%C A001333 Prime numerators with an odd index are prime RMS numbers(A140480) and
also NSW primes(A088165). [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz),
Aug 13 2008]
%C A001333 A001333(2*n+1)=A002315(n); A001333(2*n)=A001541(n). [From Ctibor O. Zizka
(ctibor.zizka(AT)seznam.cz), Aug 13 2008]
%C A001333 The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17,
58/41; essentially, numerators=A052542 and denominators=A001333.
- Clark Kimberling (ck6(AT)evansville.edu), Aug 26 2008
%C A001333 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008:
(Start)
%C A001333 Equals right border of triangle A143966. Starting (1, 3, 7,...) equals
%C A001333 INVERT transform of (1, 2, 2, 2,...) and row sums of triangle A143966.
(End)
%C A001333 Inverse binomial transform of A006012 ; Hankel transform is := [1, 2,
0, 0, 0, 0, 0, 0, 0, ...] . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Dec 04 2008]
%C A001333 Contribution from Charlie Marion (charliemath(AT)optonline.net), Jan
07 2009: (Start)
%C A001333 In general, denominators, a(k,n) and numerators, b(k,n), of continued
%C A001333 fraction convergents to sqrt((k+1)/k) may be found as follows:
%C A001333 let a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2)
%C A001333 and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);
%C A001333 let b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2)
%C A001333 and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).
%C A001333 For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5,
%C A001333 17/12, 41/29.
%C A001333 In general, if a(k,n) and b(k,n) are the denominators and numerators,
%C A001333 respectively, of continued fraction convergents to sqrt((k+1)/k)
%C A001333 as defined above, then
%C A001333 k*a(k,2n)^2-a(k,2n-1)*a(k,2n+1)=k=k*a(k,2n-2)*a(k,2n)-a(k,2n-1)^2 and
%C A001333 b(k,2n-1)*b(k,2n+1)-k*b(k,2n)^2=k+1=b(k,2n-1)^2-k*b(k,2n-2)*b(k,2n);
%C A001333 for example, if k=1 and n=3, then b(1,n)=a(n+1) and
%C A001333 1*a(1,6)^2-a(1,5)*a(1,7)=1*169^2-70*408=1;
%C A001333 1*a(1,4)*a(1,6)-a(1,5)^2=1*29*169-70^2=1;
%C A001333 b(1,5)*b(1,7)-1*b(1,6)^2=99*577-1*239^2=2;
%C A001333 b(1,5)^2-1*b(1,4)*b(1,6)=99^2-1*41*239=2.
%C A001333 Cf. A000129, A142238-A142239, A153313-153318.
%C A001333 [From Charlie Marion (charliemath(AT)optonline.net), Jan 07 2009]
%C A001333 (End)
%C A001333 Equals row sums of triangle A160756 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
May 25 2009]
%D A001333 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A001333 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover,
pp. 122-125, 1964.
%D A001333 F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint,
2006.
%D A001333 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see
p. 16.
%D A001333 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart.,
7 (1969), 231-242, see Ex. 1, pp. 237-238.
%D A001333 Reinhardt Euler, The Fibonacci Number of a Grid Graph and a New Class
of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005),
Article 05.2.6.
%D A001333 David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms,
Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
%D A001333 A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of
some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics
VI (Armidale 1978), Lect. Notes Math. 748, 1979.
%D A001333 Y. Kong, Ligand binding on ladder lattices, Biophysical Chemistry, Vol.
81 (1999), pp. 7-21.
%D A001333 Kin Y. Li, Math Problem Book I, 2001, p. 24, Problem 159
%D A001333 Munarini, Emanuele, Combinatorial properties of the antichains of a garland.
Integers, 9 (2009), 353-374.
%D A001333 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers.
2nd ed., Wiley, NY, 1966, p. 102, Problem 10.
%D A001333 H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci
Quarterly, 20 (1982), 16-21.
%D A001333 B. S. Rao, Heptagonal numbers in the associated Pell sequence ..., Fib.
Quarterly, 43 (2005), 302-306.
%D A001333 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
%D A001333 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001333 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001333 R. P. Stanley, Enumerative Combinatorics, Volume 1 (1986), p. 203, Example
4.1.2.
%D A001333 A. Tarn, Approximations to certain square roots and the series of numbers
connected therewith, Mathematical Questions and Solutions from the
Educational Times, 1 (1916), 8-12.
%D A001333 Gy. Tasi et al., Quantum algebraic-combinatoric study of the conformational
properties of n-alkanes. II, J. Math. Chemistry, 27, 2000, 191-199
(see p. 193).
%D A001333 V. Thebault, Concerning two classes of remarkable perfect square pairs,
Amer. Math. Monthly, 56 (1949), 443-448.
%D A001333 R. C. Tilley et al., The cell growth problem for filaments, Proc. Louisiana
Conf. Combinatorics, ed. R. C. Mullin et al., Baton Rouge, 1970,
310-339.
%H A001333 T. D. Noe, <a href="b001333.txt">Table of n, a(n) for n = 0..500</a>
%H A001333 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A001333 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/
poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02,
Melbourne, 2002.
%H A001333 Nick Hobson, <a href="a001333.py.txt">Python program for this sequence</
a>
%H A001333 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=143">
Encyclopedia of Combinatorial Structures 143</a>
%H A001333 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A001333 Kin Y. Li, <a href="http://www.math.ust.hk/faculty/makyli/190_2006Su/
problembk.ps">Problem 159</a>
%H A001333 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 A001333 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 A001333 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SquareRoot.html">Link to a section of The World of Mathematics.</
a>
%H A001333 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PythagorassConstant.html">Pythagoras's Constant</a>
%H A001333 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SquareTriangularNumber.html">Square Triangular Number</a>
%H A001333 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001333 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%H A001333 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A001333 a(n) = A055642(A125058(n)). - Reinhard Zumkeller, Feb 02 2007
%F A001333 a(n) = 2a(n-1) + a(n-2); a(n) = ( (1-Sqrt[ 2 ])^n + (1+Sqrt[ 2 ])^n)
/2.
%F A001333 G.f.: (1-x)/(1-2*x-x^2).
%F A001333 A000129(2n) = 2*A000129(n)*a(n). - John McNamara, Oct 30, 2002
%F A001333 a(n) = ((-i)^n)*T(n, i), with T(n, x) Chebyshev's polynomials of the
first kind A053120 and i^2 = -1.
%F A001333 a(n)=a(n-1)+A052542(n-1), n>1. a(n)/A052542(n) converges to sqrt(1/2).
- Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
%F A001333 E.g.f.: exp(x)cosh(x*sqrt(2)) - Paul Barry (pbarry(AT)wit.ie), May 08
2003
%F A001333 a(n)=sum{k=0..floor(n/2), C(n, 2k)2^k } - Paul Barry (pbarry(AT)wit.ie),
May 13 2003
%F A001333 For n >0, a(n )^2 - (1 + (-1)^(n ))/2 =sum_{k=0...n-1}((2k+1)*A001653(n-1-k));
e.g. 17^2-1=288=1*169+3*29+5*5+7*1; 7^2=49=1*29+3*5+5*1 - Charlie
Marion (charliem(AT)bestweb.net), Jul 18 2003
%F A001333 A001333(n+2) = A078343(n+1) + A048654(n) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Jan 19 2005
%F A001333 Conjecture: For prime p, A001333(p) congruent to 1 mod p ( compare with
similar comment for A000032 ) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Oct 11 2005
%F A001333 a(n) = A000129(n)+A000129(n-1) = A001109(n)/A000129(n) = sqrt(A001110(n)/
A000129(n)^2) = ceiling(sqrt(A001108(n))) - Henry Bottomley (se16(AT)btinternet.com),
Apr 18 2000
%F A001333 Also the first differences of A000129 (the Pell numbers) because A052937(n)
= A000129(n+1)+1 - Graeme McRae (g_m(AT)mcraefamily.com), Aug 03
2006
%F A001333 a(n) = Sum_{k,0<=k<=n}A122542(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Oct 08 2006
%F A001333 For another recurrence see A000129.
%F A001333 Starting (1, 3, 7, 17, 41,...), = row sums of triangle A135837. - Gary
W. Adamson (qntmpkt(AT)yahoo.com), Dec 01 2007
%F A001333 a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*2^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Dec 26 2007
%F A001333 a(n) = upper left and lower right terms of [1,1; 2,1]^n - Gary W. Adamson
(qntmpkt(AT)yahoo.com), Mar 12 2008
%F A001333 (a)=((sqrt2*(1+sqrt2)^n)-(-sqrt2*(1-sqrt2)^n))/sqrt8. Offset 0. a(3)=7
[From Al Hakanson (hawkuu(AT)gmail.com), Apr 11 2009]
%e A001333 Convergents are 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/
985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782,
... = A001333/A000129
%e A001333 The 15 3 X 2 crossword grids, with white squares represented by an o:
%e A001333 ooo ooo ooo ooo ooo ooo ooo oo. o.o .oo o.. .o. ..o oo. .oo
%e A001333 ooo oo. o.o .oo o.. .o. ..o ooo ooo ooo ooo ooo ooo .oo oo.
%p A001333 A001333 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else
2*A001333(n-1)+A001333(n-2) fi end; # version 1
%p A001333 Digits := 50; A001333 := n-> round((1/2)*(1+sqrt(2))^n); # version 2
%p A001333 with(numtheory): cf := cfrac (sqrt(2),1000): [seq(nthnumer(cf,i), i=0..50)];
# version 3
%p A001333 with(numtheory):cf := cfrac (sin(Pi/4),100): seq(nthdenom (cf,i), i=0..29
); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2007
%p A001333 A001333:=-(z+1)/(-1+2*z+z**2); [S. Plouffe in his 1992 dissertation.]
%p A001333 (Maple) a := proc(n) local M; M := (Matrix([[2,1],[1,0]])^n); M[2,1]+M[2,
2] end; seq (a(n), n=0..29); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de),
Aug 01 2008]
%t A001333 Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2],
n]]], {n, 1, 40}], 1, 1] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com),
Apr 08 2006
%t A001333 f[n_] := ((1 - Sqrt[2])^n + (1 + Sqrt[2])^n)/2; Table[Simplify@ f@n,
{n, 0, 29}] (* Or *)
%t A001333 a[0] = 1; a[1] = 1; a[n_] := a[n] = 2a[n - 1] + a[n - 2]; Table[a@n,
{n, 0, 29}] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 02 2006)
%t A001333 a=0;b=1;c=0;lst={b};Do[c=a+b+c;AppendTo[lst,c];a=b;b=c,{n,5!}];lst [From
Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 23 2009]
%o A001333 (PARI) a(n)=if(n<0,0,contfracpnqn(vector(n,i,1+(i>1)))[1,1])
%o A001333 sage: from sage.combinat.sloane_functions import recur_gen2 sage: it
= recur_gen2(1,1,2,1) sage: [it.next() for i in range(30)] - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 24 2008
%o A001333 (Other) sage: [lucas_number2(n,2,-1)/2 for n in xrange(0, 30)]# [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%o A001333 (PARI) { default(realprecision, 2000); for (n=0, 4000, a=contfracpnqn(vector(n,
i, 1+(i>1)))[1, 1]; if (a > 10^(10^3 - 6), break); write("b001333.txt",
n, " ", a); ); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net),
Jun 12 2009]
%Y A001333 For denominators see A000129.
%Y A001333 See A040000 for the continued fraction expansion of sqrt(2).
%Y A001333 a(n)+a(n+1) = 2 A000129(n+1). 2*a(n) = A002203(n) (companion Pell numbers).
%Y A001333 See also A078057 which is the same sequence without the initial 1.
%Y A001333 Cf. also A152113.
%Y A001333 Row sums of unsigned Chebyshev T-triangle A053120. a(n)= A054458(n, 0)
(first column of convolution triangle).
%Y A001333 Essentially the same as A078057. Equals A034182(n-1) + 2 and A084128(n)/
2^n. First differences of A052937. Partial sums of A052542. Pairwise
sums of A048624. Bisection of A002965.
%Y A001333 The following sequences (and others) belong to the same family: A001333,
A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533,
A002532, A083098, A083099, A083100, A015519.
%Y A001333 Second row of the array in A135597.
%Y A001333 Cf. A135837, A140750, A143966.
%Y A001333 A160756 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2009]
%Y A001333 Sequence in context: A131056 A077851 A089737 this_sequence A078057 A123335
A089742
%Y A001333 Adjacent sequences: A001330 A001331 A001332 this_sequence A001334 A001335
A001336
%K A001333 nonn,cofr,easy,core,nice,frac
%O A001333 0,3
%A A001333 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001333 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de),
Jan 10 2003
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