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A040000 a(0)=1, a(n)=2, n >= 1. 83
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Continued fraction expansion of sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))).

Inverse binomial transform of Mersenne numbers A000225(n+1)=2^(n+1)-1. - Paul Barry, Feb 28 2003

A Chebyshev transform of 2^n : if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)). - Paul Barry, Oct 31 2004

An inverse Catalan transform of A068875 under the mapping g(x)->g(x(1-x)). A068875 can be retrieved using the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. A040000 and A068875 may be described as a Catalan pair. - Paul Barry, Nov 14 2004

Sequence of electron arrangement in the 1s 2s and 3s atomic subshells. Cf. A001105, A016825. - Jeremy Gardiner, Dec 19 2004

Binomial transform of A165326. - Philippe Deléham, Sep 16 2009

Let m=2. We observe that a(n)=sum{C(m,n-2*k),k=0..floor(n/2)). Then there is a link with A113311 and A115291: it is the same formula with respectively m=3 and m=4. We can generalize this result with the sequence whose G.f is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009

With offset 1: number of permutations where |p(i)-p(i+1)|<=1 for n=1,2,..,n-1.  This is the identical permutation and (for n>1) its reversal.

Equals INVERT transform of bar(1, 1, -1, -1,...).

Eventual period is (2). - Zak Seidov, Mar 05 2011

Also decimal expansion of 11/90. - Vincenzo Librandi, Sep 24 2011

a(n) = 3 - A054977(n); right edge of the triangle in A182579. - Reinhard Zumkeller, May 07 2012

REFERENCES

A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000

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

MathPath Square-roots via Continued Fractions - Mats Granvik, Jul 18 2009

Eric Weisstein's World of Mathematics, Square root.

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

G. Xiao, Contfrac

Index entries for continued fractions for constants

Index to divisibility sequences

Index to sequences with linear recurrences with constant coefficients, signature (1).

FORMULA

G.f.: (1+x)/(1-x). - Paul Barry, Feb 28 2003

a(n) = 2-0^n; a(n) = sum{k=0..n, binomial(1, k)}. - Paul Barry, Oct 16 2004

a(n) = n*sum{k=0..floor(n/2), (-1)^k*binomial(n-k, k)*2^(n-2k)/(n-k)}. - Paul Barry, Oct 31 2004

A040000(n) = sum{k=0..floor(n/2), C(n-k, k)(-1)^k*A068875(n-k)}. - Paul Barry, Nov 14 2004

Euler transform of length 2 sequence [2, -1]. - Michael Somos, Apr 16 2007

G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-v)*(u+v) -2*v* (u-w). - Michael Somos, Apr 16 2007

E.g.f.: 2*exp(x) - 1. - Michael Somos, Apr 16 2007

a(-n) = a(n). - Michael Somos, Apr 16 2007

G.f.: (1-x^2)/(1-x)^2. - Jaume Oliver Lafont, Mar 26 2009

G.f.: exp(2*atanh(x)). - Jaume Oliver Lafont, Oct 20 2009

a(n) = sum_{k=0..n} A108561(n,k)*(-1)^k. - Philippe Deléham, Nov 17 2013

a(n) = 1 + sign(n). - Wesley Ivan Hurt, Apr 16 2014

EXAMPLE

sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))). - Harry J. Smith, Apr 21 2009

MAPLE

Digits := 100: convert(evalf(sqrt(2)), confrac, 90, 'cvgts'):

MATHEMATICA

ContinuedFraction[Sqrt[2], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

PROG

(PARI) {a(n)= 2-!n} /* Michael Somos, Apr 16 2007 */

(PARI) a(n)=1+sign(n). - Jaume Oliver Lafont, Mar 26 2009

(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(2)); for (n=0, 20000, write("b040000.txt", n, " ", x[n+1])); } [Harry J. Smith, Apr 21 2009]

(Haskell)

a040000 0 = 1; a040000 n = 2

a040000_list = 1 : repeat 2  -- Reinhard Zumkeller, May 07 2012

CROSSREFS

Convolution square is A008574.

Cf. A001333/A000129.

See A003945 etc. for (1+x)/(1-k*x).

From Jaume Oliver Lafont, Mar 26 2009: (Start)

Equals A000012(n)+A000012(n-1).

Sum_{0<=k<=n} a(k) = A005408(n).

Prod_{0<=k<=n} a(k) = A000079(n). (End)

Cf. A113311, A115291, A171418, A171440, A171441, A171442, A171443.

Cf. A000674 (boustrophedon transform).

Sequence in context: A046698 A007395 A036453 * A239374 A055642 A138902

Adjacent sequences:  A039997 A039998 A039999 * A040001 A040002 A040003

KEYWORD

nonn,cofr,easy

AUTHOR

N. J. A. Sloane, Dec 11 1999

STATUS

approved

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Last modified December 20 03:43 EST 2014. Contains 252241 sequences.