A040000 a(0)=1; a(n)=2 for n >= 1.

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

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_{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

Also, number of self-avoiding walks and coordination sequence for the one-dimensional lattice Z. - Sean A. Irvine, Jul 27 2020

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.

Bruce Fang, Pamela E. Harris, Brian M. Kamau, and David Wang, Vacillating parking functions, arXiv:2402.02538 [math.CO], 2024.

Kshitij Education, Molar specific heat

MathPath, Square-roots via Continued Fractions

Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, arXiv:2309.04012 [math.NT], 2023.

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.

Index entries for 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-2*k)/(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

Example

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

Maple

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

Mathematica

ContinuedFraction[Sqrt[2], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
a[ n_] := 2 - Boole[n == 0]; (* Michael Somos, Dec 28 2014 *)

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.

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

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

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. A001333/A000129 (continued fraction convergents).

Cf. A000122, A002193 (sqrt(2) decimal expansion), A006487 (Egyptian fraction).

Cf. 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: A211662 A007395 A036453 * A211665 A239374 A262190

Adjacent sequences: A039997 A039998 A039999 * A040001 A040002 A040003

Keyword

nonn,cofr,easy,cons

Author

N. J. A. Sloane, Dec 11 1999

Status

approved