A074294 Integers 1 to 2*k followed by integers 1 to 2*k + 2 and so on.

1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 2

Offset

1,2

Comments

From Cino Hilliard, Sep 13 2004: (Start)

Also the numerator of the fraction in the continued fraction expansion of sqrt(n) for nonsquare n = 2,3,5,6,7... . E.g., for n = 7,

sqrt(7).=.2.+._3_................

...............4..+._3_..........

.....................4..+._3_....

...........................4.....

3 is the 5th entry in the table. sqrt(1) and sqrt(4) are not included because 1 and 4 are squares." (End)

A074294 is the natural fractal sequence of A002061; the corresponding natural interspersion is A194011; see A194029 for definitions. - Clark Kimberling, Aug 17 2011

It appears that this is also a triangle read by rows in which row n lists the first 2*n positive integers, n >= 1 (see example). - Omar E. Pol, May 29 2012

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Formula

a(n) = n - 2*binomial(floor(1/2 + sqrt(n)), 2).

a(n^2 + n) = 2*n.

a(n) = n - 2 - floor(sqrt(n)+3/2)*floor(sqrt(n)-3/2). - Mikael Aaltonen, Jan 02 2015

G.f.: x/(1-x)^2 - (2*x/(1-x))*sum(k>=1, k*x^(k^2+k)). That sum is related to Jacobi theta functions. - Robert Israel, Jan 05 2015

a(n) = n + A000194(n) - A053187(n). - Robert Israel, Jan 05 2015

Example

From Omar E. Pol, May 29 2012: (Start)
Written as a triangle the sequence begins:
1, 2;
1, 2, 3, 4;
1, 2, 3, 4, 5, 6;
1, 2, 3, 4, 5, 6, 7, 8;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
Row n has length 2*n = A005843(n). (End)

Maple

seq(seq((j-n^2-n), j=n^2+n+1..(n+1)^2+n+1), n=0..20); # Robert Israel, Jan 05 2015

Mathematica

A074294[n_] := n - 2*Binomial[Floor[1/2 + Sqrt[n]], 2] (* Enrique Pérez Herrero, Apr 14 2010 *)
Table[Range[2n], {n, 10}]//Flatten (* Harvey P. Dale, Oct 20 2018 *)

Prog

(PARI) {a(n) = n - 2 * binomial( floor( 1/2 + sqrt(n)), 2)}
(PARI) c(n) = for(x=2, n, if(issquare(x)==0, a=floor(sqrt(x)); print1(x-a^2", "))) /* Cino Hilliard, Sep 13 2004 */
(Haskell)
import Data.List (inits)
a074294 n = a074294_list !! (n-1)
a074294_list = f $ inits [1..] where
   f (xs:_:xss) = xs ++ f xss
-- Reinhard Zumkeller, Apr 14 2014

Crossrefs

Cf. A002061, A194011.

Cf. A071797.

Cf. A000194, A053187.

Sequence in context: A327189 A255045 A194103 * A168265 A062050 A358086

Adjacent sequences: A074291 A074292 A074293 * A074295 A074296 A074297

Keyword

nonn,easy,tabf

Author

Michael Somos, Aug 20 2002

Status

approved