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A001333 Numerators of continued fraction convergents to sqrt(2).
(Formerly M2665 N1064)
278
1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243, 275807, 665857, 1607521, 3880899, 9369319, 22619537, 54608393, 131836323, 318281039, 768398401, 1855077841, 4478554083, 10812186007, 26102926097, 63018038201, 152139002499 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of n-step non-selfintersecting paths starting at (0,0) with steps of types (1,0), (-1,0) or (0,1) [Stanley].

Number of n steps one-sided prudent walks with east, west and north steps. - Shanzhen Gao, Apr 26 2011

Number of ternary strings of length n-1 with subwords (0,2) and (2,0) not allowed. - Olivier Gérard, Aug 28 2012

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

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.

Number of (n-1) X 2 binary arrays with a path of adjacent 1's from top row to bottom row. - R. H. Hardin, Mar 16 2002

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.

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).

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.

Binomial transform of A077957. - Paul Barry, Feb 25 2003

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, Jun 02 2004

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, Aug 06 2004

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, Dec 18 2004

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

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, Mar 22 2003 [Amended by Paul E. Black (paul.black(AT)nist.gov), Dec 18 2006]

a(n) mod 10 = A131707. See A131711. - Paul Curtz, Apr 08 2008

Prime numerators with an odd index are prime RMS numbers(A140480) and also NSW primes(A088165). - Ctibor O. Zizka, Aug 13 2008

The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17, 58/41; essentially, numerators=A052542 and denominators here. - Clark Kimberling, Aug 26 2008

Equals right border of triangle A143966. Starting (1, 3, 7,...) equals INVERT transform of (1, 2, 2, 2,...) and row sums of triangle A143966. - Gary W. Adamson, Sep 06 2008

Inverse binomial transform of A006012 ; Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

Contribution from Charlie Marion, Jan 07 2009: (Start)

In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:

let a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2) and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);

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) and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).

For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5, 17/12, 41/29.

In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then

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

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);

for example, if k=1 and n=3, then b(1,n)=a(n+1) and

1*a(1,6)^2-a(1,5)*a(1,7)=1*169^2-70*408=1;

1*a(1,4)*a(1,6)-a(1,5)^2=1*29*169-70^2=1;

b(1,5)*b(1,7)-1*b(1,6)^2=99*577-1*239^2=2;

b(1,5)^2-1*b(1,4)*b(1,6)=99^2-1*41*239=2.

Cf. A000129, A142238-A142239, A153313-A153318.

(End)

This sequence occurs in the lower bound of the order of the set of equivalent resistances of n equal resistors combined in series and in parallel (A048211). - Sameen Ahmed Khan, Jun 28 2010

Let M = a triangle with the Fibonacci series in each column, but the leftmost column is shifted upwards one row. A001333 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 27 2010

a(n) is the number of compositions of n when there are 1 type of 1 and 2 types of other natural numbers. - Milan Janjic, Aug 13 2010

Equals the INVERTi transform of A055099. - Gary W. Adamson, Aug 14 2010

From L. Edson Jeffery, Apr 04 2011: (Start)

Let U be the unit-primitive matrix (see [Jeffery])

U=U_(8,2)=

(0 0 1 0)

(0 1 0 1)

(1 0 2 0)

(0 2 0 1).

Then a(n)=(1/4)*Trace(U^n). (See also A084130, A006012.)

(End)

For n>=1, row sums of triangle

m/k.|..0.....1.....2.....3.....4.....5.....6.....7

==================================================

.0..|..1

.1..|..1.....2

.2..|..1.....2.....4

.3..|..1.....4.....4.....8

.4..|..1.....4....12.....8....16

.5..|..1.....6....12....32....16....32

.6..|..1.....6....24....32....80....32....64

.7..|..1.....8....24....80....80...192....64...128

which is triangle for numbers 2^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012

a(n) is also the number of ways to place k non-attacking wazirs on a 2 X n board, summed over all k>=0 (a wazir is a leaper [0,1]). - Vaclav Kotesovec, May 08 2012

The sequences a(n) and b(n) := A000129(n) are entries of

  powers of the special case of the Brahmagupta Matrix - for details see Suryanarayan's paper. Further, as Suryanarayan remark, if we set A = 2*(a(n)+b(n))*b(n), B = a(n)*(a(n)+2*b(n)), C = a(n)^2+2*a(n)*b(n)+2*b(n)^2 we obtain integral solutions of the Pythagorean relation A^2 + B^2 = C^2, where A and B are consecutive integers. - Roman Witula, Jul 28 2012

Pisano period lengths:  1, 1, 8, 4, 12, 8, 6, 4, 24, 12, 24, 8, 28, 6, 24, 8, 16, 24, 40, 12,.... - R. J. Mathar, Aug 10 2012

A001333 and A000129 give the diagonal numbers described by Theon of Smyrna. - Sture Sjöstedt, Oct 20 2012

a(n) is the top left entry of the n-th power of any of the following six 3X3 binary matrices: [1, 1, 1; 1, 1, 1; 1, 0, 0] or [1, 1, 1; 1, 1, 0; 1, 1, 0] or [1, 1, 1; 1, 0, 1; 1, 1, 0] or [1, 1, 1; 1, 1, 0; 1, 0, 1] or [1, 1, 1; 1, 0, 1; 1, 0, 1] or [1, 1, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

REFERENCES

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

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.

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.

S. Gao, H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).

David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.

Martin Griffiths, Pell identities via a quadratic field, International Journal of Mathematical Education in Science and Technology, 2013, DOI:10.1080/0020739X.2013.790512.

R. P. Grimaldi, Ternary strings ..., Congressus Numerantium, 205 (2011), 129-149.

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.

Y. Kong, Ligand binding on ladder lattices, Biophysical Chemistry, Vol. 81 (1999), pp. 7-21.

Kuba, Markus; Panholzer, Alois. Enumeration formulae for pattern restricted Stirling permutations. Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From N. J. A. Sloane, Sep 25 2012

J. V. Leyendekkers and A. G. Shannon, Pellian sequence relationships among pi, e, sqrt(2), Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62; http://www.nntdm.net/papers/nntdm-18/NNTDM-18-2-58-62.pdf. See Table 2. - From N. J. A. Sloane, Dec 23 2012

Kin Y. Li, Math Problem Book I, 2001, p. 24, Problem 159

Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 102, Problem 10.

H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20 (1982), 16-21.

B. S. Rao, Heptagonal numbers in the associated Pell sequence ..., Fib. Quarterly, 43 (2005), 302-306.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.

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, Volume 1 (1986), p. 203, Example 4.1.2.

E. R. Suryanarayan, The Brahmagupta Polynomials,  Fibonacci Quarterly, 34.1 (1996), 30-39.

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.

Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see Eq. (21)).

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).

V. Thebault, Concerning two classes of remarkable perfect square pairs, Amer. Math. Monthly, 56 (1949), 443-448.

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..500

Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), (2000) 175-179. [From Sameen Ahmed Khan, Jun 28 2010]

Joerg Arndt, Matters Computational (The Fxtbook), pp.313-315

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

J. Bodeen, S. Butler, T. Kim, X. Sun, S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7

Fan Chung, Ron Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194

K. Dohmen, Closed-form expansions for the universal edge elimination polynomial, arXiv preprint arXiv:1403.0969, 2014

Nick Hobson, Python program for this sequence

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 143

L. E. Jeffery, Unit-primitive matrices)

Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346 [From Sameen Ahmed Khan, Jun 28 2010]

Tanya Khovanova, Recursive Sequences

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 392-3.

Dmitry Kruchinin, Integer properties of a composition of exponential generating functions, arXiv:1211.2100

Kin Y. Li, Problem 159

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.

Eric Weisstein's World of Mathematics, Square Root

Eric Weisstein's World of Mathematics, Pythagoras's Constant

Eric Weisstein's World of Mathematics, Square Triangular Number

Index entries for "core" sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for sequences related to linear recurrences with constant coefficients, signature (2,1)

FORMULA

a(n) = A055642(A125058(n)). - Reinhard Zumkeller, Feb 02 2007

a(n) = 2a(n-1) + a(n-2);

a(n) = ((1-sqrt(2))^n + (1+sqrt(2))^n)/2.

G.f.: (1 - x) / (1 - 2*x - x^2) = 1 / (1 - x / (1 - 2*x / (1 + x))). - Michael Somos, Sep 02 2012

A000129(2n) = 2*A000129(n)*a(n). - John McNamara, Oct 30 2002

a(n) = (-i)^n *T(n, i), with T(n, x) Chebyshev's polynomials of the first kind A053120 and i^2 = -1.

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

E.g.f.: exp(x)cosh(x*sqrt(2)). - Paul Barry, May 08 2003

a(n) = sum{k=0..floor(n/2), C(n, 2k)2^k }. - Paul Barry, May 13 2003

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, Jul 18 2003

a(n+2) = A078343(n+1) + A048654(n). - Creighton Dement, Jan 19 2005

Conjecture: For prime p, a(p) congruent to 1 mod p (compare with similar comment for A000032). - Creighton Dement, Oct 11 2005

a(n) = A000129(n)+A000129(n-1) = A001109(n)/A000129(n) = sqrt(A001110(n)/A000129(n)^2) = ceiling(sqrt(A001108(n))). - Henry Bottomley, Apr 18 2000

Also the first differences of A000129 (the Pell numbers) because A052937(n) = A000129(n+1)+1. - Graeme McRae, Aug 03 2006

a(n) = Sum_{k,0<=k<=n} A122542(n,k). - Philippe Deléham, Oct 08 2006

For another recurrence see A000129.

a(n) = Sum_{k, 0<=k<=n}A098158(n,k)*2^(n-k). - Philippe Deléham, Dec 26 2007

a(n) = upper left and lower right terms of [1,1; 2,1]^n. - Gary W. Adamson, Mar 12 2008

E.g.f.: exp(x)*cosh(x*sqrt(2)) = G(0)/2; G(k)=(-1)^k+1/(((3+sqrt(2))^k)-x*(1+sqrt(2))*((17+12*sqrt(2))^k)/(x*(1+sqrt(2))*((3+sqrt(2))^k)-(k+1)/G(k+1)) ;  (continued fraction). - Sergei N. Gladkovskii, Dec 04 2011

For n>=2, a(n)=F_n(2)+F_(n+1)(2), where F_n(x) is Fibonacci polynomial (cf.A049310): F_n(x)=sum{i=0,...,floor((n-1)/2)}C(n-i-1,i)x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012

a(-n) = (-1)^n * a(n). - Michael Somos, Sep 02 2012

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

G.f.: (1+G(0))/(4*x), where G(k)= x*(2*k-1) - 1 + 4*x + x*(2*k-1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013

EXAMPLE

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.

The 15 3 X 2 crossword grids, with white squares represented by an o:

ooo ooo ooo ooo ooo ooo ooo oo. o.o .oo o.. .o. ..o oo. .oo

ooo oo. o.o .oo o.. .o. ..o ooo ooo ooo ooo ooo ooo .oo oo.

If p[1]=1, and 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)=det A. - Milan Janjic, Apr 29 2010

1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 41*x^5 + 99*x^6 + 239*x^7 + 577*x^8 + ...

MAPLE

A001333 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else 2*procname(n-1)+procname(n-2) fi end;

Digits := 50; A001333 := n-> round((1/2)*(1+sqrt(2))^n);

with(numtheory): cf := cfrac (sqrt(2), 1000): [seq(nthnumer(cf, i), i=0..50)];

A001333:=-(z+1)/(-1+2*z+z**2); # Simon Plouffe in his 1992 dissertation

a:= proc(n) local M; M := (Matrix([[2, 1], [1, 0]])^n); M[2, 1]+M[2, 2] end: seq (a(n), n=0..30); # Alois P. Heinz, Aug 01 2008

MATHEMATICA

Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]], {n, 1, 40}], 1, 1] - Stefan Steinerberger, Apr 08 2006

f[n_] := ((1 - Sqrt[2])^n + (1 + Sqrt[2])^n)/2; Table[Simplify@ f@n, {n, 0, 29}] (* Or *)

a[0] = 1; a[1] = 1; a[n_] := a[n] = 2a[n - 1] + a[n - 2]; Table[a@n, {n, 0, 29}] (* Or *)

Table[ MatrixPower[{{1, 2}, {1, 1}}, n][[1, 1]], {n, 0, 30}] (from Robert G. Wilson v, May 02 2006)

a=c=0; t={b=1}; Do[c=a+b+c; AppendTo[t, c]; a=b; b=c, {n, 40}]; t (* or *) LinearRecurrence[{2, 1}, {1, 1}, 40] (* Vladimir Joseph Stephan Orlovsky, Mar 23 2009 *)

Join[{1}, Numerator[Convergents[Sqrt[2], 30]]] (* Harvey P. Dale, Aug 22 2011 *)

PROG

(PARI) {a(n) = if( n<0, (-1)^n, 1) * contfracpnqn( vector( abs(n), i, 1 + (i>1))) [1, 1]} /* Michael Somos, Sep 02 2012 */

(PARI) {a(n) = polchebyshev(n, 1, I) / I^n} /* Michael Somos, Sep 02 2012 */

(Sage) from sage.combinat.sloane_functions import recur_gen2

it = recur_gen2(1, 1, 2, 1)

[it.next() for i in range(30)] ## Zerinvary Lajos, Jun 24 2008

(Sage) [lucas_number2(n, 2, -1)/2 for n in xrange(0, 30)] # Zerinvary Lajos, Apr 30 2009

(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); ); } \\ Harry J. Smith, Jun 12 2009

(Haskell)

a001333 n = a001333_list !! n

a001333_list = 1 : 1 : zipWith (+)

                       a001333_list (map (* 2) $ tail a001333_list)

-- Reinhard Zumkeller, Jul 08 2012

CROSSREFS

For denominators see A000129.

See A040000 for the continued fraction expansion of sqrt(2).

a(n)+a(n+1) = 2 A000129(n+1). 2*a(n) = A002203(n) (companion Pell numbers).

See also A078057 which is the same sequence without the initial 1.

Cf. also A152113.

Row sums of unsigned Chebyshev T-triangle A053120. a(n)= A054458(n, 0) (first column of convolution triangle).

Row sums of A140750, A160756, A135837.

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.

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

Second row of the array in A135597.

A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176501, A176502. - Sameen Ahmed Khan, Jun 28 2010

Cf. A055099. - Gary W. Adamson, Aug 14 2010

Cf. A028859, A001906 / A088305, A033303, A000225, A095263, A003945, A006356, A002478, A214260, A001911 and A000217 for other restricted ternary words. - Olivier Gérard, Aug 28 2012

Sequence in context: A077851 A089737 * A123335 A078057 A089742 A187258

Adjacent sequences:  A001330 A001331 A001332 * A001334 A001335 A001336

KEYWORD

nonn,cofr,easy,core,nice,frac,changed

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

STATUS

approved

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Last modified September 18 17:06 EDT 2014. Contains 246924 sequences.