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A040000
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a(0)=1; a(n)=2 for n >= 1.
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152
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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;
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refs;
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text;
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OFFSET
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0,2
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COMMENTS
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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, 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_{k=0..floor(n/2)} binomial(m,n-2*k). 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
With offset 1: minimum cardinality of the range of a periodic sequence with (least) period n. Of course the range's maximum cardinality for a purely periodic sequence with (least) period n is n. - Rick L. Shepherd, Dec 08 2014
With offset 1: n*a(1) + (n-1)*a(2) + ... + 2*a(n-1) + a(n) = n^2. - Warren Breslow, Dec 12 2014
With offset 1: decimal expansion of gamma(4) = 11/9 where gamma(n) = Cp(n)/Cv(n) is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Sep 11 2016
a(n) equals the number of binary sequences of length n where no two consecutive terms differ. Also equals the number of binary sequences of length n where no two consecutive terms are the same. - David Nacin, May 31 2017
a(n) is the period of the continued fractions for sqrt((n+2)/(n+1)) and sqrt((n+1)/(n+2)). - A.H.M. Smeets, Dec 05 2017
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REFERENCES
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A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
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LINKS
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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.
Kshitij Education, Molar specific heat [Broken link]
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
Wikipedia, Poisson's constant
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for eventually constant sequences
Index to divisibility sequences.
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FORMULA
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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)} binomial(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) for all n in Z (one possible extension to n<0). - 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
10 * 11/90 = 11/9 = (11/2 R)/(9/2 R) = Cp(4)/Cv(4) = A272005/A272004, with R = A081822 (or A070064). - Natan Arie Consigli, Sep 11 2016
a(n) = A001227(A000040(n+1)). - Omar E. Pol, Feb 28 2018
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EXAMPLE
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sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))). - Harry J. Smith, Apr 21 2009
G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + ...
11/90 = 0.1222222222222222222... - Natan Arie Consigli, Sep 11 2016
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MAPLE
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Digits := 100: convert(evalf(sqrt(2)), confrac, 90, 'cvgts'):
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MATHEMATICA
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ContinuedFraction[Sqrt[2], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
a[ n_] := 2 - Boole[n == 0]; (* Michael Somos, Dec 28 2014 *)
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PROG
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(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
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CROSSREFS
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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).
Cf. A000122.
Other continued fractions for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040002 (contfrac(sqrt(5)) = (2,4,4,...)), A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).
Sequence in context: A046698 A007395 A036453 * A239374 A262190 A055642
Adjacent sequences: A039997 A039998 A039999 * A040001 A040002 A040003
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KEYWORD
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nonn,cofr,easy,cons
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AUTHOR
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N. J. A. Sloane, Dec 11 1999
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STATUS
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approved
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