0,2

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 (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Nov 14 2004

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

Binomial transform of A165326. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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 generalise this result with the sequence whose G.f is given by (1+z)^(m-1)/(1-z) Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 08 2009

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

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

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

MathPath Square-roots via Continued Fractions [From Mats Granvik (mats.granvik(AT)abo.fi), Jul 18 2009]

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

G. Xiao, Contfrac

Index entries for continued fractions for constants

Index entries for sequences related to linear recurrences with constant coefficients

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

a(n)=2-0^n; a(n)=sum{k=0..n, binomial(1, k)}. - Paul Barry (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Oct 31 2004

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

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 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 26 2009]

G.f.: exp(2*atanh(x)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 20 2009]

sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]

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

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

(PARI) a(n)=1+sign(n) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 26 2009]

(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]

Convolution square is A008574.

Cf. A001333/A000129.

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

Contribution from Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 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)

nonn,cofr,easy,new

N. J. A. Sloane (njas(AT)research.att.com).