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Search: id:A040000
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| A040000 |
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a(0)=1, a(n)=2, n >= 1. |
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+0
37
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| 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; listen)
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OFFSET
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0,2
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COMMENT
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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]
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REFERENCES
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A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,20000
Index entries for sequences related to linear recurrences with constant coefficients
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
MathPath Square-roots via Continued Fractions [From Mats Granvik (mats.granvik(AT)abo.fi), Jul 18 2009]
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FORMULA
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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]
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EXAMPLE
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sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
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MAPLE
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Digits := 100: convert(evalf(sqrt(2)), confrac, 90, 'cvgts'):
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PROGRAM
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(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]
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CROSSREFS
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Convolution square is A008574.
Cf. A001333/A000129.
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)
Sequence in context: A130130 A046698 A036453 this_sequence A007395 A055642 A138902
Adjacent sequences: A039997 A039998 A039999 this_sequence A040001 A040002 A040003
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KEYWORD
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nonn,cofr,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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