%I A040000
%S A040000 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,
%T A040000 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,
%U A040000 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
%N A040000 a(0)=1, a(n)=2, n >= 1.
%C A040000 Continued fraction expansion of sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))).
%C A040000 Inverse binomial transform of Mersenne numbers A000225(n+1)=2^(n+1)-1. 
               - Paul Barry (pbarry(AT)wit.ie), Feb 28 2003
%C A040000 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 (pbarry(AT)wit.ie), 
               Oct 31 2004
%C A040000 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 (pbarry(AT)wit.ie), Nov 14 2004
%C A040000 Sequence of electron arrangement in the 1s 2s and 3s atomic subshells. 
               Cf. A001105, A016825. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), 
               Dec 19 2004
%C A040000 Binomial transform of A165326. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 16 2009]
%D A040000 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A040000 A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
%H A040000 Harry J. Smith, <a href="b040000.txt">Table of n, a(n) for n=0,...,20000</
               a>
%H A040000 MathPath <a href="http://www.mathpath.org/Algor/squareroot/algor.square.root.contfrac.htm">
               Square-roots via Continued Fractions</a> [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Jul 18 2009]
%H A040000 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 A040000 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PythagorassConstant.html">Pythagoras's Constant</a>
%H A040000 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">
               Contfrac</a>
%H A040000 <a href="Sindx_Con.html#confC">Index entries for continued fractions 
               for constants</a>
%H A040000 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A040000 G.f.: (1+x)/(1-x) - Paul Barry (pbarry(AT)wit.ie), Feb 28 2003
%F A040000 a(n)=2-0^n; a(n)=sum{k=0..n, binomial(1, k)}. - Paul Barry (pbarry(AT)wit.ie), 
               Oct 16 2004
%F A040000 a(n)=n*sum{k=0..floor(n/2), (-1)^k*binomial(n-k, k)*2^(n-2k)/(n-k)}. 
               - Paul Barry (pbarry(AT)wit.ie), Oct 31 2004
%F A040000 A040000(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*A068875(n-k)}. - Paul 
               Barry (pbarry(AT)wit.ie), Nov 14 2004
%F A040000 Euler transform of length 2 sequence [ 2, -1]. - Michael Somos Apr 16 
               2007
%F A040000 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
%F A040000 E.g.f.: 2*exp(x) - 1. - Michael Somos Apr 16 2007
%F A040000 a(-n) = a(n). - Michael Somos Apr 16 2007
%F A040000 G.f.: (1-x^2)/(1-x)^2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Mar 26 2009]
%F A040000 G.f.: exp(2*atanh(x)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Oct 20 2009]
%e A040000 sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))) 
               [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
%p A040000 Digits := 100: convert(evalf(sqrt(2)),confrac,90,'cvgts'):
%o A040000 (PARI) {a(n)= 2-!n} /* Michael Somos Apr 16 2007 */
%o A040000 (PARI) a(n)=1+sign(n) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Mar 26 2009]
%o A040000 (PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(2)); 
               for (n=0, 20000, write("b040000.txt", n, " ", x[n+1])); } [From Harry 
               J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
%Y A040000 Convolution square is A008574.
%Y A040000 Cf. A001333/A000129.
%Y A040000 See A003945 etc. for (1+x)/(1-k*x).
%Y A040000 Contribution from Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 
               26 2009: (Start)
%Y A040000 Equals A000012(n)+A000012(n-1)
%Y A040000 Sum_{0<=k<=n}a(k) = A005408(n)
%Y A040000 Prod_{0<=k<=n}a(k) = A000079(n)
%Y A040000 (End)
%Y A040000 Sequence in context: A130130 A046698 A036453 this_sequence A007395 A055642 
               A138902
%Y A040000 Adjacent sequences: A039997 A039998 A039999 this_sequence A040001 A040002 
               A040003
%K A040000 nonn,cofr,easy
%O A040000 0,2
%A A040000 N. J. A. Sloane (njas(AT)research.att.com).